I don't claim to be an expert, but my take is this:
Schwartz's defense was designed to maximize the playmaking of the front 4 and allow the rest of the defense to play with 7 DBs in various coverages as much as possible. The DL was focused on penetration from wide positions with lots of stunting. The secondary used more zones and focused on disguise to allow the DL time to work. The basic idea is that if you can stop the run with 7 and get pressure with four, you are going to be very tough to move the ball against.
Rex's scheme is to play much more man coverage to allow more players in the box- safeties and LBs to play the run and blitz (or fake blitz). The DL plays more traditional run first defense and have more two gap responsibilities. The defense is really designed to work with superior linebackers (especially the outside guys), excellent cornerbacks and big immovable linemen. I thin Rex has been playing much more 1 gap with the DL than he did in NY but they are still no where near as free to wreak havoc as they were under Schwartz.
One thing that has improved- with the DL playing more conservatively and more players in the box the run defense is better than last year. However, the pass defense relies on confusing the blocking schemes and effective blitzing and that hasn't been happening.
Simply put, I think that Schwartz's scheme may fit the Bills personnel better but they are talented enough as a group to play any decent scheme and be successful.