I've played quite a few games with this gambit, must've caught a couple dozen people out with it but as everyone else has said, 4. Nxd4, 4.c3, or 4.0-0 leave white ahead in development and therefore with a slight edge. The game is still playable for black of course but it's not a solid opening, just one I use because I dislike the Italian game.
Game 2568194 is what black is aiming for, although any line after 4.Nxe5 Qg5 will give black an edge.
I wouldn't really call this opening a gambit - more like a trap. A gamble on the possibility of white taking the pawn.
If for example white plays:
4. c3 Nxf3
5. Qxf3...
- white has 2 pieces developed, black has 0.
- white defends e4.
- white is ready to play d4
- white is ready to castle
- white threatens Qxf7, limiting Blacks replies.
Other 4th moves for white yield similar advantages:
4. O-O (now Nxe5 is threatened because g2 is defended)
4. d3 (Defending g5, see above)
however, I'm not a fan of the 4. Nxd4 option, but it's probably marginally better for white.
As long as white continues with normal development, and doesn't fall into the trap - he should be better off than black.