This is the first Critical Position in the Jerome Gambit. Although 99% of the time (according to my database) Black captures the Bishop, the move is not forced. It is simply a choice for the second player between having an objectively winning game (4...Kxf7) and having the worse position (after 4...Kf8/4...Ke7 5.Bb3). Yet a few defenders will adopt the sly attitude If he wants me to take the Bishop, then I won't take it. There is some "psychology" in this, too: the Jerome Gambiteer suddenly finds himself or herself "stuck" with a calm, but immesurably "better" game than had been expected a move before. This change of fortune can take some getting used to. Those who resist may wish to resort to 5.Bxg8 or 5.Qe2 (if 5...Kxf7 6.Qc4+); or transpose to the Evans Jerome Gambit with 5.b4.
4...Kxf7 This is the second Critical Position. The "classical" Jerome Gambit continues with 5.Nxe5 (about 84% of the games in my database) while "modern" Jerome Gambits continue with alternatives such as 5.Nc3, 5.d3, or 5.0-0. For the record, after a very long think (over 12 hours) Deep Rybka 3.0 Aquarium assesses Black as being 1.91 pawns better after 5.Nxe5+ as well as after 5.Nc3. It sees White being only 1.72 pawns worse after 5.d3 or 5.0-0.
One fifth of a pawn doesn't seem like a lot to me, and I still prefer the complications of 5.Nxe5+. Jerome Gambiteers who feel they can knuckle down and simply outplay their opponents with the "modern" variations are free to disagree with me.5.Nxe5+
This is the third Critical Position. Black's most frequent response is the logical 5...Nxe5, but he has alternatives in 5...Kf8, 5...Ke8 and 5...Ke7. The first of these "others" is a respected defense going back to Jerome - Brownson, Iowa 1875 (1/2-1/2, 29). The other two are blunders that give White the advantage after 6.Qh5.
After 5.Nxe5+ Kf8, the Banks Variation (see "Jerome Gambit and Vlad Tepes..." and "Jerome Gambit, Vlad Tepes... and Garlic!") is tempting, but probably Alonzo Wheeler Jerome's 6.Nxc6 is best.
5...Nxe5 This is the fourth Critical Position. White most frequently follows up with 6.Qh5+, although 6.d4 is also played, and there are a number of rare other moves.