What are the differences between a seatpost-mounted and frame-mounted rear rack
The advantages of seat mounted racks are that they  don’t require rear dropout rack lugs; some of the seat mount racks have quick release levers so you can easily take them off (while racing or transporting the bike, etc), and are many times the only option for a rear suspension bike. 

The disadvantages of seat mount racks are numerous:


max luggage weight is limited
many regular panniers that require a bottom (dropout) hook won’t fit
some are designed only for proprietary bags and bag mounts
they are often heavier (thx @michael)
minimum seat drop becomes limited 
you can’t use many seat droppers and suspension seatposts as there isn’t enough clearance
seat post damage is possible
doesn’t fit some oval, rhomboidal, or oversized seat posts
if you have a round seatpost and even a modest weight on the back and you take a spill or even do a modest spin-turn, it can be enough for the rack to spin the seat post


Source: had a Specialized hardrock hardtail back in the day and had a regular rack on it. Switched over to a seat mounted rack at one point as I thought it looked cooler and it does, but quickly realized it was most just looks.