Assuming we're talking about PCI-E cards here, I'm going to guess Computer A is newer and Computer B is older, or at least the motherboards are.
Computer A probably is running PCI-E 2.0 while Computer B is PCI-E 1.1
Video Card B is probably newer, it's probably PCI-E 2.0 which won't run on PCI-E 1.1 compliant motherboards. PCI-E 2.0 came out around Nvidia 8xxx cards I believe.
Video card A is probably older so it's probably PCI-E 1.1, which works in both 1.1 and 2.0 motherboards.