What's the difference between 128-bit and 256-bit VRAM?

That bit width is nothing to do with the ram, it is the bus connection between the ram and the gpu and it does not make it run faster.

It does allow more data at once to be transferred at the same time and allows more data per second overall given the ram is the same in both cases.

It may or may not make a difference depending on game and settings but it is not given that a graphics card with a higher bus width will be faster.

You cannot make any fast and hard rules which has really been the case since ati have supported memory types other than gddr3.

Look up reviews for either the card you are looking for or any equivalent before purchasing, do not look at specs.

 
Indeed. What actually matters is memory bandwidth which is a combination of bus size and memory speed. For example cards with DDR5 ram and a 128 bit bus actually have slightly more bandwidth than those with DDR3 and a 256 bit bus.
In general the average person looking to purchase a video should really ignore almost all the individual specs of a card(apart from different versions of the same card.) What matters in the end will be performance so benchmarks are all that really matters.