I feel like people tend to downplay the differences betwern the two versions. Here are a few that I immediately noticed while playing:

-Classic's undo system isn't "pillow dependent" like in full body where you are only limited to about 3 consecutive undos at a time, where you can add more if you collect more pillows. In Classic you can undo a lot more times. -The pillows in Classic are for "retries." In full body you have unlimited retries. -Full body introduces a new main chcracter called Rin. Slight spoiler, but they'll help you as you go along the game. As they're not in Classic, there won't be anyone helping you when you're in trouble. -With a new main character comes new endings not found in the Classic version. -Full body also has more anime cutscenes. Personally they really helped flesh out Catherine and Vincent's relationship more, and these tidbits were quite enjoyable to watch. The Classic version doesn't have these cutscenes but it still manages to stand on its own. -More characters (not main) are added in Full Body. There's a particular pair whose friendship really had me amused all throughout and had me smiling by the end of the game. -Full body has a remix option primarily for returning players which kind of alters the levels in which there are new special blocks. However, there's also the option to play them in their original form. -This is purely a nitpick, but one of the drinks in the bar (whiskey) is replaced with wine. It does not affect the gameplay whatsoever but...it's different.

Whichever version you choose, I highly recommend playing both if you enjoyed your experience. You probably won't get this now but don't worry about things getting too repetitive because the liquor trivia is different for each version (thanks Atlus :D). I feel as though Classic has a bit darker, more serious tone to it while Full Body has a lighter tone.