Probably the strongest move, but White can also play immediately 13.Rxe7!? with an improved version of Torre's sacrifice against Santasiere in the Dimock Theme Tournament. Play should continue 13...hxg5! (13....Nxe7 14.Bxf6 gxf6 15.Qxh6 should be winning for White) 14.Nxg5 Nxe7 15.Nce4 Ned5! 16.Rxd5 Re8 17.Rd3 +=.
This move seems to lose, as the sequel should have demonstrated. But it is difficult to find a satisfactory improvement. On 13....Qd7, for example, White has the familiar sacrifice 14.Rxe7! leading to a clear advantage. And after 13....d5, to defeat 14.Rxe7, White instead sacrifices the Bishop with 14.Bxh6!
Here are some lines to show the power of White's attack even against Black's best alternatives:
1) 13....d5 14.Bxh6 gxh6 15.Qxh6 Qd6 16.Ng5 Qf4+ 17.Kb1 Qxf2 18.Red1 Qxg2 19.Nxd5 +- or 18....Qf5 19.Rh3 and all roads lead to mate.
2) 13...Qd7 14.Rxe7! Nxe7 (14....Qxe7 15.Nd5 Qe6? 16.Nxf6+ gxf6 17.Bxf6 gives White a mating attack while 15....Nxd5 16.Bxe7 += is at least better for White) 15.Bxf6 gxf6 (15....Ng6 16.Ne5! += is a familiar tactic in this variation, giving White a clearly superior ending in all variations) 16.Ne4! Qe6 17.Nd4 Qe5 18.Rh3 Ng6 19.Nxf6+ Kg7 20.Qxh6+ Kxf6 21.Rf3+ Ke7 22.Re3 += and though material is relatively balanced (Queen and pawn versus two Rooks), White retains a strong initiative since the Queen and Knight cooperate very well here to create continued threats against Black's King.
3) 13...Kh7!? (a strange attempt at defense, but one of the more difficult to defeat) 14.Bxh6! (White really must take up the challenge, since 14.Bxf6!? Bxf6 15.Qe4+ Kg8 16.Nd5 Be5 is not very clear) 14...gxh6 15.Ng5+ Kg6! (Not quite as good is 15...Kg7 16.Rg3 Ng4 17.Rxe7!! Qxe7 18.Rxg4 f5 19.Ne6+ Kf7 20.Rg7+Kxe6 21.Rxe7+ Nxe7 += or perhaps 20.Qxh6!? fxg4 21.Nd5! += or 20...Rg8 21.Rxg8 Rxg8 22.Qh7+ Kxe6 23.Qxg8+ Kd7 24.Kd1 += and White has a clear edge in all these lines) 16.Rg3 Ng4 (Not 16...Nh5 17.Rxe7!! [less clear is 17.Qe4+!? f5 18.Qe6+ Bf6 19.Nf7+ Nxg3 20.Nxd8 Nxd8 and Black has a lot of material for the Queen] 17....Nxg3 18.Ne6 Ne2+ 19.Kd2 Qxe7 20.Qg4+ Qg5+ 21.Nxg5 hxg5 22.Nxe2 +=) 17.Qxg4! hxg5 18.Nd5 Ne5 19.Qe2! with a powerful attack for White.
More complicated, but ultimately more powerful, is 18.g4! winning the Knight at h5. Black can then initiate a counter-attack (which Browne's 18.Re3 is intended to counter), but it does not work out in the end after 18....Qe1+ 19.Nd1 Nd4 20.c3 Ne2+ 21.Kc2 Nef4 22.Re3 Qf1 23.gxh5 Ng2 24.Qg3 Nxe3+ 25.Qxe3 and White has regained the intiative with a winning material advantage of two Knights versus a Rook.
It is not clear why White offered a draw here in a winning position. He is a pawn up with the better placed pieces and the initiative to boot. This is the ideal outcome of his attack. Black will certainly be able to organize some counterplay on the queenside eventually, but White surely retains the edge with good play. One can only speculate that personal reasons, unrelated to the final position, made it necessary for White to terminate the game.