The first Neo game that hooked me was the original Fatal Fury.
There was a big red in the 7-11 near my school at the time it came out. I was already big into SFII, which was clearly the better game, but the FF character designs, music, and story elements had me dropping quarters until I beat it many times. Andy's design/moveset plus Geese falling out that window are the things that really hooked me.
My roommate at the time was buying Gamest magazine every month, so he had been reading all about Fatal Fury 2 a while before it came out and was super-hyped about it. Then it showed up in a big red in a Japanese market near the apartment, and we were both blown away. We'd already been playing the first AOF on that same cab quite a bit, though SFII at a nearby 7-11 was still our main game. Until Fatal Fury 2.
About one week of dropping mass amounts of money into FF2, and we decided to combine resources and buy a homecart system. Street Fighter basically became a thing of the past. We got a slightly used homecart system, extra old style joystick, used FF1 & AoF1 carts, and FF2 new. That pretty much cleaned both of us out for months. And we played FF2...and later, FFS...more or less non-stop most hours of the day or night.
So I'd say Fatal Fury was the first game to hook me, and Fatal Fury 2 was the point of no return.