The following game shows how gambiteers would love the Jerome Gambit to proceed. White's play is thematic and aggressive, while Black's errors tend to be reasonable-looking moves that do not stand up to the situation they are played in. (I should point out that the time limit for the game was several days per move.)
obviously - vallabhan
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+
4...Kxf7 5.Nxe5+ Nxe5 6.Qh5+ Ke6 7.Qf5+ Kd6 8.f4 Ng6
Elsewhere I have referred to this move as "somewhat inexact", as it saves one threatened piece while allowing another piece to go by the board. See "Reliable".
Stronger alternatives are the proactive 8...Kc6 and the counterattacking 8...Qh4+.
9.Qd5+ Ke7 10.Qxc5+ d6 11.Qf2
Putting the Queen on the f-file, where it can be backed up by a Rook on f1, and threaten the enemy King. It also increases the tactical complexity of the position by not guarding the pawn on e4.
I am amazed that this move hasn't been repeated, according to The (55,560 games) Database, as it is the essence of Jerome-ness.
The alternate move 12.d3 has 4 wins and 3 losses.
If Black plays the logical 12...Nxe4, then Stockfish 8's main line recommendation is 13.Qf3 Ng5 14.Qg3 Ne4 15.Qf3 with a draw by repetition - which may make sense to a computer, but probably not to a human.
I would be more inclined to answer 12...Nxe4 with 13.Qe2, when after 13...Bxf5 14.O-O Qd7 15.d3 Rae8 16.dxe4 Kd8 17.Nc3 Be6 the game is even, but not over.
This removes the Knight from danger, and puts pressure on White's d4 pawn. Nonetheless, cranky Stockfish 8 prefers 13...Nf7. (That move seems a bit defensive, and I am not sure that a human would be thinking "defense" a piece up - but maybe he should.)
14.Nc3 Re8 15.Bg5 Kf8 16. O-O-O h6 17. Bh4 Bd7
White's moves flow, and the position screams "pawn storm on the Kingside". It doesn't happen, but only because White breaks up Black's Kingside instead.
Getting out of the pin of the Knight on f6, and getting off the possibly soon-to-be hot d-file; but Black needed to do something drastic, like 18...Kg8 19.Nd5 Rf8 20.e5 Nxd5!? 21.Bxd8 Rxf5 22.Qg3 Nxd8 23.exd6 cxd6 24.Qxd6 Bc6, when he would have three pieces for his Queen, but White should still be better. (Yes, I had help in figuring this out.)
19.Bxf6 gxf6 20.Qh4 Kg7 21.Nd5 Qd8
Black is in a lot of trouble on the Kingside. The better way of defending his f-pawn, 21...Rf8, still leads to problems, for example 22.Qg4+ Kh8 23.Qg6 Bxf5!? 24.exf5 Qd7 25.Qxh6+ Qh7. Returning the extra piece has allowed Black's Queen to participate in the defense of his King, but White has two extra pawns - and his Rooks are bound to cause trouble as they move up and over to the g- and h-files.
Yes, White's e-pawn cannot capture the Bishop without giving up the Rook, but that was never the plan.
23.Rg3+ Kh8 24.Qxh6+ Bh7 25.Qg7 checkmate