OMG so hard to choose because am a MASSIVE fan of both, but capcom takes it for the creation of Ryu, the ultimate martial artist and archtype of the 2d genre hands down.
SNK is very fast, the king of fighters series was great at first but progressively got worst and worst, after 96 it became nigh unplayable and coined the legendary SNK bosses were they would kill you with 1 hit and be far superior in all aspects etc. The Fatal Fury series brough us the legendary Terry Bogard and Joe Higashi, both extremely cool and well, theres ofcourse Art of Fighting. Samuria Showdown was crap though, and your man in the avatar Haohmaru is just a complete tool lol, though its cool when he throws the katana in the air then catches it in the sabath

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Capcom launched the greatest fighting game in history with Street Fighter II, so good it was reincarnated 10 times and had a winning formula that was copied over again and again. Its characters are pretty universally known and who wouldnt wanna spar with Chun Li? Ah chunners.....
The alpha series was ok, they released other good series like Darkstalkers and Rival Schools though they didnt take off. But Capcom also expanded their game types and now are a strong company in japan, where as SNK died off then got bought over by some flunky organisation just to sell shoddy products through their distinguishable label.
So hard to call, but id go 60/40 in Capcoms favour.