What is the advantage of Ne7 instead of Nf6 in this or other related openings?

I like to play the c or f pawns and then advance the knight behind (both black and white) at certain times. Here I don't see the point, and looking at a Stockfish line following it through, I still don't see the point in Ne7. What would you have done in this early opening position? Nf6, Ne7, something else?

There are 2 points:

1. On e7 the knight can go to either g6 or (after ...c5) to c6. On f6 it can only go to h5, loses a pawn to Nxe5 and Qxh5 or to d7, blocking the diagonal of the important Bc8.
2. On e7 it allows the move ...f6 to support e5. This is most clear in the Derdl variation 1 e4 e5 2 Nf3 Nc6 3 Bb5 a6 4 Ba4 Nf6 5 O-O Be7 6 Bxc6 dxc6. White deliberately loses a tempo Bb5-Ba4-Bxc6, because Nf6 and Be7 are misplaced: should be Bd6 and f6.

Nf6 looks fine to me. It cannot be bad. I think Ne7 would need a lot of exact analysis, especially if going for f6.