How? Well, I'd guess at Forge ID Resolver. I strongly suspect it only resolves conflicts when the config files generate, which for the most part is equivalent to "when the mod is first installed".
So if on one computer you install mod A first and then mod B, and on the other computer you install mod B first and then mod A (and run minecraft between each install, so that the older mod gets a chance to write its config), then you could potentially get a situation where the first computer would've given priority to mod A and shuffled B's block IDs around, and on the second computer the opposite happened and A's block IDs were shuffled.
The easiest solution is not letting Forge ID Resolver have anything to do in the first place -- once you have working config files on one computer, copy them to the other one, which ensures both have the same block IDs and other configuration settings.
All of this is pure guesswork, of course, I'm not even sure that you really mean "block IDs" when you say "slots".