Playing Ne4 instead of Be4 was the losing move, yes. The only other thing I could think of is that your bishop was on a good diagonal before and on move 14 it might have been interesting to recapture d3 with the queen instead of the bishop, would have kept your opponent on the defensive longer, maybe.
Where did I go wrong? English opening

I agree with Renacer: 12...Be4 was a tactical mistake. However, it looks like you started to go wrong even earlier with 8...Nb4, which moves the same piece twice in the opening. In my experience, Black is often struggling to keep up with White in development in the opening, especially in a d4-opening (which is very similar to many c4-openings), so much so that even trading a pawn (11...dxc4 here) is something of which I am leery in such openings. I usually try to let White do all the exchanges while I rush to develop as Black, while I watch for mistakes. I've found that often the first "mistake" White makes is to leave Black the option of ...Bb4+ that forces a trade of bishops (...Bb4+ Bd2 ...Bxd2+), which eases Black's game considerably, and pushes the game toward an equalized draw. But if White *continues* to make mistakes after that, then I will push to relieve him of his incompetent monarch.

Again for a long time your moves looked quite allright. Instead of 15...Rc8, would probably exchange with 15...BxR 16BxR etc and white has nothing. Earlier white's play looked weird but can't see where you can get winning advantage. (At least not without switching my "helper" on). After 15...Rc8 he could get two bishops with NxB, with some advantage maybe.
Actually my helper informs me 15...BxR 16BxR black can win a pawn with 16...Bxc4. It's not +1, so it sees some compensation, for a human not clear what that is. As it happened nb4, didn't acheive much, and white could have kicked it at somepoint with a3. It's not exactly a bad move though.

I thought 15. ...B*R 16. B*Ra8 Q*B Q*B then white would get a chance to do d4 so to avoid it and to get Rook at open file if d4 c*d4 I brought my rook to Rc8. Don't you think that seems more logical?
Apart from Ne4 move tell me if I missed out anything or how should I have played. Please post your analysis.