5.Bxf7+ has a reasonable idea behind it, in that it exposes Black's king and jettisons White's bishop before Black can use the fork trick on it that would follow with 5.Nxe4 d5. However, it has some concrete problems:
- After 5...Kxf7 6.Nxe4, Black plays 6...d5 and forms a massive center.
- Black's king turns out to not be vulnerable on f7. It will quickly retreat to g8 where it is relatively safe.
- White's knights are going to get kicked around a lot with loss of tempo. If he plays 7.Neg5+ Kg8, both ...h6 and ...e4 will attack knights that don't have very good places to flee to.
In contrast, after 5.Nxe4 d5, Black regains his sacrificed piece, but has to break up his pawn center in the process. After 6.Bd3 dxe4 7.Bxe4, White no longer has his opening advantage but the position is relatively equal.
The lesson from all of this is that 4.Bc4 is not a very good move. The standard alternatives are 4.Bb5 (Spanish Four Knights), 4.d4 (Scotch Four Knights), and 4.g3 (Glek Four Knights), which all keep White's opening advantage.
P.S. I should point out that there is a third alternative for White's fifth move, 5.0-0, gambiting a pawn for a lead in development. This is likely actually better than either 5.Bxf7+ or 5.Nxe4, but if you want to play a gambit like that then there are much better ones to play than the Italian Four Knights.