How do I compare seemingly equal wheelsets?
The Hed wheels may be slightly better in an objective sense, but the absolute difference may be small. Additionally, as a consumer, it may be hard to tell where they're better, and by how much, just from the specs. As outlined below, a lot of differences may appear in areas that consumers can't see on their own.
You might get a general sense of quality by asking around on internet forums or in bike stores, but these responses will inevitably be prone to the responder's own confirmation bias, placebo effects, and other subjective issues. I recall a podcast (Marginal Gains) where the author (Josh Poertner, who's worked at Zipp wheels before) relayed a story: he was working with the US Postal Service team, and the riders complained that his carbon wheels were too flexy (that is, they flexed side to side when ridden hard). Josh had tested the wheels' lateral stiffness, and they objectively measured up to competitor wheels. He got in touch with an aluminum anodizing company, and asked them to anodize a batch of the hubs for the wheels in a very bright red. He swapped out the hubs ... and lo and behold, everyone said the wheels were now very stiff and fast. Most responses on the internet will be plagued by issues like this, and the fact is that the Hed wheels could be prone to this because they have a very nice finish (see below). This response tries to outline objective factors that might differ from rim to rim in general.
I’m not able to see a price attached to the Specialized wheels, and I suspect they’re OEM wheels (I.e. they come stock on a lot of Specialized bikes, and aren’t usually sold aftermarket). I suspect that a $250 price might just be for a front wheel, but I could be wrong. Either way, the Hed premium may or may not be worth it.
Tolerances - rim, hubs, and build quality
All components have some manufacturing tolerance. For example, a manufacturer's site might say that X rim weighs 450g. In reality, that means the mean weight is 450g, with some standard deviation or variance around the mean. Good rims will vary less. Poor quality rims will vary more. The rims can be more or less round. Relevant to tubeless tires, their internal diameter may be on the high side of the industry-specified tolerance, or it may be right on the tolerance. (Side note: aluminum rims are extruded from a die, but the die wears as time goes by. Hence, a rim's mean weight actually creeps up over time. All manufacturers are likely tempted to understate their mean weight, but perhaps the poorer ones might have a greater incentive to do this.)
I have not patronized November Bicycles, which builds custom wheels, but I trust their analysis. They have spoken very highly of the tolerances and overall quality of Hed rims in general. Here’s their review of the Eroicas.
In contrast, the Specialized wheels are likely to have a rim from a generic Taiwanese manufacturer, e.g. Kinlin or Alex. By itself, this is not a detriment. In fact, if you go through November’s blog, you’ll see they’ve spoken favorably of many Kinlin rims, and they’ve offered some in wheel builds. Many companies selling wheels are in fact using some combination of Taiwanese rims and hubs. In fact, I've had wheels with Kinlin rims and generic Taiwanese hubs before, and I've been quite satisfied. The potential issue with the Specialized wheels is that we don’t know what rims and what hubs. Their rims might be lower quality and higher variance in various tolerances than the Heds. In practice, this might mean that if you buy a pair of Specialized wheels, the probability that the rim wasn’t as round as it should be is higher than on the Heds - albeit the chance of an issue should still be low, and we consumers can't estimate that chance reliably. The same issue with manufacturing tolerances applies to hubs.
The issue with tolerances also applies to the build quality - that is, how true and round (I.e. lateral and vertical tolerance after building) the finished wheels are, and how even the spoke tension is. This affects the wheel’s lifespan. Most mass-produced wheels are machine laced, and finished by hand, and the build quality varies. The issue is, there’s no reliable way for us to estimate that tolerance for Hed and Specialized wheels. I note that you said the Hed wheels are built by hand. In principle, I think that a skilled wheel builder can achieve better results than the usual mass production technique, but I don’t necessarily know that what the average skill level is at Hed, and I can’t confirm from their site that all wheels are built by hand.
Some other places to look for differences between wheels are as follows.
Spokes
Type
The Specialized site says that they use DT Competitions, which are 14/15 gauge, or 2.0mm/1.8mm butted spokes. The Hed site isn’t informative, but they said round steel spokes, and they’re very likely to be similar to the Competitions. It’s possible that Hed used a super light butted spoke (e.g. DT Revolution or Sapim Laser, both 14/17g, where 17g is 1.4mm), but they would have said. Many wheels at the Hed price point use a bladed spoke, which results in a small aerodynamic advantage (November bicycles calculated the advantage at 11 grams of drag at 30mph, which is about 1 watt; comparison was a 38mm carbon rim with 20 Sapim CX-rays, which are 14/17g bladed spokes, vs the same rim, 20 Sapim Lasers, 14/17g double butted round). Wheels at lower price points may use straight gauge spokes (e.g. 14 gauge, or 2mm, through their whole length). These are heavier, and possibly less durable, than butted spokes.
Count
The Specialized wheels are 24 spokes front and rear, and the Heds appear to be equal. Heavier riders may want 28 spokes on the rear, or both front and rear. Fewer spokes are lighter and more aerodynamic, but the differences should be small.
Other spoke and related characteristics
Related to the spokes, many wheels use alloy nipples. These are a minor savings over brass nipples. Alloy nipples have a slightly higher chance of corroding and breaking, particularly if ridden in poor weather. It's also possible to round out an alloy nipple when truing the wheel. With the best quality alloy nipples, I believe that these issues are minimal. An issue might present if you're on a wheel that uses poorer quality alloy nipples. Otherwise, brass is a durable choice, and will penalize you by a mere 20g per wheel (or less).
Rim characteristics
Internal width
Another point of differentiation you need to pay attention to is the rim internal width. The Eroica is designed for big tubeless tires. I have no personal experience yet, but I think you would need at least a 28mm tire on that rim, and the Eroica is really a gravel rim. The Specialized rim is more typical in width, and can support road or gravel tires. Also see what November wrote about the Eroica’s bead lip, which is designed to improve tubeless tire retention.
Side note: for someone buying aerodynamic carbon wheels, you want to make sure that your tire width is a bit less than the rim's external width. If you want to run relatively wide tires (e.g. 25mm or 28mm nominal width, which may stretch to 28-30mm when mounted and inflated on a 23mm+ internal width rim), then you will incur an aerodynamic penalty if you exceed the rim's width. The average user might quite fairly ask, what do I care? However, if you've gone and bought an aerodynamic carbon wheelset, then you do want to at least consider this - you've gone and offset some of the aerodynamic benefit you just bought, so you may not want to do this unless you've got a good reason (e.g. the race course has some gravel sectors that are better handled on 28mm tires).
Depth
In general, rim depth may be a point of differentiation. Both these wheels are roughly equal in depth. In general, alloy rims tend to go up to about 30mm deep. Generally, the deeper rim, the more aerodynamic the wheel is. However, between 20 and 30mm deep, I believe the aerodynamic differences are small.
Construction
Cheaper rims may use a cheaper means of construction, e.g. a sleeved joint on many entry-level wheels versus a welded joint on more premium wheels, maybe even a pinned joint on a really cheap wheel. There's some discussion on rim joining methods here. NB that it's possible to produce high-quality pinned wheels. Cheaper rims may not be as smooth at the rim joint, so if those are rim brake wheels, the rider might feel a pulsing sensation each time the brakes pass the joint. Altenatively, cheaper rims might have the sidewalls more poorly machined, so rim braking doesn't feel as smooth. Obviously, this doesn't apply to disc brake wheels like the ones in the original post.
Last, more expensive wheels may use better alloys. I recall that aluminum alloys with niobium were slightly lighter and stronger than some other alloys - although, in 2017, niobium alloys were used by only one Taiwanese manufacturer, Kinlin. There are likely some points of differentiation between other aluminum alloys. However, I'm not familiar with metallurgy, so I can't provide further detail. In principle, if we were talking carbon wheels, similar differences would exist between both the carbon fibers and the resins used. Again, I'm not familiar enough with carbon wheel construction to give an opinion here.
Tubeless issues
Both rims in the original post are certified for tubeless use. However, at the time of writing, tubeless industry standards are still evolving. Not all tires may play perfectly with all rims. Reading November's blog, Hed's other rims seem to have good general compatibility. It's possible that other rim brands will not play as nicely with some tubeless tires, or even that one brand might make a few rim models that don't play as well. If the wheel's manufacturing tolerances are loose, especially around rim diameter and the diameter of the center channel in the rim, it's possible that some rims of a particular model will be on the loose side of the tubeless specifications, and they won't seat tires properly.
If this happens, you could be more vulnerable to burping your tires unexpectedly - this is where one of the beads comes loose from the rim temporarily, and you lose pressure and sealant. This is not good on gravel. If you are running road tubeless, this is really bad. Road tubeless is run at high pressure, so you will lose air much faster than on a gravel tire. This can cause people to crash. At speed.
Again, this is something that's hard for consumers to know and to measure. At best, you could ask mechanics if they've had problems with a wheel or rim brand or model. One possible issue with asking about that Specialized wheelset is that in the US, many shops will mainly sell Specialized bikes, and thus they may have an incentive to play down problems with the brand. Or mechanics may simply not have had enough experience with that wheel to give you a good answer.
Finish
Last, the finish of the rims is one area of differentiation. Rims might be done with a more polished or a matte finish, but either way, the good finishes should have consistent color across the rim and from rim to rim, as well as more durable clear coating. Additionally, good finishes will just look better. November has said previously that Hed wheels tend to have very nice finishes, and that the prices for the rims and wheels reflect that! In a purely objective sense, there is no point paying a premium for finish. November has commented elsewhere that there are rims that are high quality in other aspects and that are cheaper than Heds. But not everyone makes decisions purely on objective factors, and you are not obliged to do so either.
Hubs
Some name-brand hubs are known to be high quality (e.g. White Industries, Chris King, DT Swiss, Industry9). Some of these are quite expensive, but some are relatively good value for money (e.g. DT Swiss 350 or 370, although NB these versions are made in Taiwan whereas the 240 is made in Switzerland; a number of OEM wheels use DT Swiss 350 or 370 hubs, and I suspect these tend to be good value for money). Some premium hubs are also available in pretty anodized colors, but I won't focus on that because it's a purely subjective consideration.
I can't confirm, but I suspect that both the Specialized and Hed wheels have generic Taiwanese hubs, as I alluded to above. Many Taiwanese hubs are decent, but some are lower quality. It's impossible to tell which is which just from sitting here. Better quality hubs will tend to have better tolerances in their bearing bores, and better quality bearings. They may also have wider spacing between the wheel bearings. These should lead to longer bearing life. Additionally, bearing spacing, and other geometrical aspects of the hub (e.g. spacing between flanges) may affect the stiffness of the wheel. Remember that many generic Taiwanese brands are fine; in particular, I've heard that hubs made by Novatec and Bitex are good, as are the DT Swiss hub models made in Taiwan.
Elsewhere on this site, I've suggested avoiding hubs with particularly small bearings (especially 688 bearings for front hubs), as these aren't big enough to bear the sort of loads you take on bicycles, and will quickly wear out. That doesn't appear to be an issue with the wheels in the original post.
On the other end of the price range, one might encounter hubs with ceramic bearings. Good ceramic bearings may have lower rolling resistance than steel, and ceramic bearings are frequently used in industrial applications. Many bicycle ceramic bearings also have relatively low-contact seals, so these might be vulnerable to contamination. They'll need more regular maintenance than you might otherwise do to your hubs. In addition, they may need or may benefit from special low-friction grease. Objectively, the price per watt gained here is very high, and many of us won't notice the gains here. That said, I suspect ceramic bearings are more sensible in hubs than in the bottom bracket (hubs rotate faster than your cranks).
Degrees of engagement
The Hed hubs have 3 pawls, and the Specialized hubs have 4. Chances are that you will have to turn your cranks a smaller radius before your free hub engages with the Specialized wheels. That is, the Specialized hub may have more points of engagement than the Hed.
I would not care about this unless you are riding very technical gravel. Cyclingtips has a good discussion of points of engagement here. It does objectively matter for mountain bikers. They frequently have to stop pedaling, reposition their feet, and then pedal, perhaps while on a very steep hill. In that scenario, you obviously want your freehub to engage immediately.
Roadies will not objectively benefit from this. In fact, more points of engagement may mean more drag while coasting. Many gravel riders will not benefit either, unless they're on very technical terrain. There are some gravel bikes that have geometry similar to mountain bikes and that can handle terrain approaching what MTBs can handle. The OP's wheel choice and comments suggest that he's using these wheels on a gravel bike, so this might be something to consider.
The Gestalt
Overall, wheels will vary in their claimed weight. Lighter rims, hubs, spokes, and nipples will all reduce the wheel's overall weight. Some words of caution apply, however.

Small reductions in weight will only minimally reduce your time on a
given climb.
Furthermore, manufacturers often under-state the claimed weight on a wheel. It could be that the smaller, less well-known manufacturers have more of an incentive to under-state their weights. After all, weight is easily measurable, and 100 grams sounds like a lot of weight. It is not; see the first point. You will naturally be able to feel a 100g difference between two sets of wheels when you lift them with your hands, but almost all of us are not pedaling our bikes with our hands.
It's possible to sacrifice durability if you go too lightweight on some components. For example, a manufacturer might build their hubs with small bearings that aren't sufficient to take the load. They might use an axle that is too light and gets bent in regular use. Rims that are too light can be prone to cracking.

Wheels will also vary in their lateral stiffness. In the worst case, it's possible to actually (i.e. not a placebo effect as discussed in the introduction) flex a wheel side to side when you pedal hard, such that you rub the rim on the brake pads. Stiffness is a complex phenomena affected by the rim, the geometry of the hub, and the gauge and number of the spokes.