One important difference between the Abrahams Jerome and the regular Jerome Gambit is the value of ...Bxf2+ for Black. In the latter, the move serves mostly a "psychological" purpose (You sacrificed a piece to upset my King, so I will sacrifice a piece to upset your King), as discussed in "Trading Down Against the Jerome Gambit" and earlier posts. However, in the former, the return Bishop sacrifice is often the best move available.
ndizvoh - stevebrown
blitz, FICS, 2015
1.e4 e5 2.Bc4 Bc5 3.Bxf7+ Kxf7 4.Qh5+ g6 5.Qxe5
A quick look in The Database shows 4,097 games have reached this position, with White scoring 53%. Only 1,151 games include the "best" response, 5...Bxf2+, which reduces White's scoring to only 52%.
Clearly there are other factors than the "best" move affecting the outcome!
A comparison with the Abrahams Jerome Gambit line 1.e4 e5 2.Bc4 Bc5 3.Bxf7+ Kxf7 4.Qh5+ Kf8 5.Qxe5 is even more puzzling. The Database has 5,026 games reaching this position, with White scoring 41% (suggesting that 4...Kf8 might be a stronger response than 4...g6). Yet, in the 219 games where Black makes his "best" response, 5...Bxf2+, White improves his scoring to 47%.
It's worth repeating: Clearly there are other factors than the "best" move affecting the outcome!
Black sees he will be losing his Bishop, and decides to get a pawn for it.
Declining the Bishop is rare, according to The Database, and not a good idea. One example: KevinSI - stevebrown, standard, FICS, 2014: 6.Kf1 Qf6 7.Nf3 Qxe5 8.Nxe5+ Ke6 9.Kxf2 Kxe5 10.Re1 Nf6 11.d3 Rf8 12.Kg1 Nc6 13.Nd2 Nd4 14.Nc4+ Ke6 15.Rf1 Kf7 16.Bg5 Kg7 17.c3 Ne6 18.Bxf6+ Rxf6 19.Rxf6 Kxf6 20.Rf1+ Kg7 21.Ne5 d6 22.Rf7+ Kg8 White resigned
Black protects his Bishop and Rook. He also had the move 6...Nf6, which does the same thing, as 7.Kxf2 would then be met with 7...Ng4+, winning White's Queen.
White probably should have settled for 7.Qxf6+ Nxf6 8.Kxf2 Nxe4+ and a roughly equal game.
7...Kg7 8.Nf3 Ne7 9.Qc4 Bb6
Black has retained his extra piece. White has an extra pawn, but given the lack of safety of his King, it is not adequate compensation.
10.Rf1 d6 11.Nc3 Bg4 12.Nd5 Nxd5 13.Qxd5 c6 14.Qc4 Re8 15.d3 d5 16.e5 Rxe5+ White resigned