Yes, its double the amount of bit
If your talking about the amount of RAM on the card though for example 128mb and 256mb then this would matter more with texture cache which could speed the game up, and games will need more texture cache as they are starting to now (why cards are going to come out with 512mb of memory and 1gb instead of 256).
Doom 3 for example on the Ultra Quality setting uses uncompressed textures which gives better quality of course, but to cache all the textures onto the video card, it'd need at least 512mb and so it loads it when needed instead of all together.
Yes the second question you asked is true, since the newer card would be faster, as its not all about the memory included. Its the core speed, and amount of rendering pipes too, etc. The memory would only help again about textures etc, but if you have a bad video card (the older one) then of course it won't perform as well in games as a 128mb newer card would
I hope this helps