Do LTE modules depend on specific carrier?
This question can be answered on two different levels.  There is the general answer, and the answer that applies to the device you linked.

LTE is a wireless standard and the radios are no more or less interoperable than, say, various brands of 802.11n radios are with each other.  

That is, as long as your radio supports the band you want to use.  Just like 802.11n, the same standard can be used on multiple bands.  There are only two in the case of the 802.11n standard, the 2.4GHz and 5GHz bands, which I am sure you're quite familiar with.  LTE, however, has dozens of frequency bands it can be used on.

And just like Wifi, if your router only talks on  the 5GHz band, and your wifi card can only talk on the 2.4GHz band, then despite both being 802.11n, they will be totally unable to communicate with each other.  LTE works exactly the same way.  The important part really isn't the LTE part, it's which LTE bands a given radio supports, and which bands the carrier(s) you want to use the radio with supports. 

So you absolutely cannot assume an LTE radio will necessarily work with any carrier besides the one it is specified for use with.  Always check the specs and see what bands it can use, and compare that to the list of bands used by the networks you want to use the module with.

In the specific case of the module you linked, it supports every band used by both AT&amp;T and Verizon (it also helps that those two carriers have a lot of overlap so you really only need one extra band to support both), so it will have no problems hopping onto either carrier's network, providing you have an activated SIM card inserted for the carrier you want.  

To be a bit more clear, there is no physical difference between the two modules you've linked, except for the SIM card that is included with it.  The reason it is sold like that is due to a huge pile of regulatory bureaucracy and approvals and nonsense you don't need to worry about.  

However, it should be noted that the non-LTE part, the 3G fallback, is going to be through AT&amp;T regardless.  Both modules use an HL7588, which only supports UTMS 3g.  Verizon does not have a single UTMS tower, while AT&amp;T has an abundance of them.  

Given that I could not even find in existence a module by that company that has CDMA 3G AND LTE that also worked with AT&amp;T's bands, I am fairly certain the other module, the ATT W/ VZW 3G is an error.  Both modules would be limited to AT&amp;T 3G as a fall back.  But there can be roaming provisions on the included SIM card to complicate things further.  I would ask Sierra Wireless if you care about 3G.

But, for LTE, either module can be used with either carrier, provided you have the necessary SIM card.  

You could not use it on 'other carriers besides AT&amp;T' however.  You can use it with one other carrier, Verizon.  It supports the LTE bands used by AT&amp;T and Verizon.  Those two carriers and no others.  And again, you would of course need a Verizon SIM card to use it with Verizon.