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What is the difference between these two opposite color bishop endgames?

Hello,

In the attached blitz game, apparently 40.Bxe4 wins and 40.fxe4 draws, according to tablebase. I played 40.Bxe4 and won, but I did not spend a lot of time in-game to understand the difference.
I would like to ask what is the core difference between these two endgames. I would assume that having a passed pawn promoting on a light square rather than a dark square would help white. But is the problem with 40.fxe4 that it gets the two passed pawns closer and so makes it possible for the black king to defend both somehow?

Thanks a lot!

The game:
I always said that the bishop, like the rook, should stand behind the passed pawn. On this and other tips, players always dismissed me as an annoying fly. But I can not help it - I always try to teach, although no one asks me about it.
One basic rule: the bigger the distance between the passers the better for the attacker.

Other „rules“: the more passed pawns the better, 3 vs 2 is better than 2:1, 1:0 is a certain draw. The defending bishop must try to stop on a single diagonal in order not to get „teared“.
I'd have to say: consult the tablebase! Play it out and see what the differences are; but yes, the pawns being closer together presents more drawing opportunities.
Thanks all!
@Sarg0n: All rules seem very logical, but could you elaborate on the last one please? I couldn't quite understand what you meant.
@Sarg0n : wow, this wikipedia article is priceless to me. Thank you so much! Mind if I ask about similar articles you might know of? (or better: how to find those myself)
#5 Honestly, having read multiple books and basically every online article about opposite-colored bishops, I found it extraordinarily difficult to believe that 40. fxe4?? draws. But as I try numerous variations against the tablebase, each time I find Black putting his bishop on the b8-h2 diagonal and White unable to promote either pawn.

Move everything one file to the right and Black's defensive task is either more difficult or lost after gxf4 - White can threaten to play Kb6-Ka7-Kb8 then c7-c8=Q.
Indeed, it is really difficult to play otb. By the way, I played yesterday the following with white, 5+0 rated. The analysis around move 43 is somewhat faulty because the position is constatntly winning according to the TB.

At normal circumtames , there has to be at least 3 file for win This endgame between pawns . fxe4 , there is 2 file between pawns , D raw , ( even hard to hold as black ) Bxe4 is win Because there is 3 file between pawns , therefore should be win .

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