capcom vs. snk, and sega vs. namco. close competition in the same genres, a rich exchange of ideas, and as i understand it lots of headhunting. of course these were arcade rivalries, and arcades are dead now.
SNK vs CAPCOM (for arcade fighters)
SEGA vs Nintendo (Old School rivalry)
SEGA's AM2 vs Tecmo's Team Ninja (more like, whatever AM2 does, Team Ninja copies)
NAMCO vs SEGA (arcade rivalry and brethren)
Midway vs Acclaim (back when it used to matter)
NAMCO vs Nintendo (they used to hate each other)
Square vs Nintendo (see above)
Atari vs Nintendo (bitterness & jealousy begot Tengen, which begot lawsuit)
Sony vs Nintendo
Nintendo's R&D1 vs EAD
Konami vs Nintendo (for top Japanese publisher)