First off, Congrats on getting a LEV! You helped beat/break the system!
The difference is interesting. Feel Better Soup heals 50% of your Max HP and also blocks full air when it is used. LEV simply gives you a FULL HP recovery in battle. But that's all. Though if you use it outside of the dome, it's used up and vanishes forever. Poof! So if you decide to equip the LEV, make sure you Equip it, and not Use it when doing so.
Both of these healers can be used Once Per Battle.
I hope that helps!