Like said above it is just the amount of VRAM. Usually how it works is the performance gap between a 512MB card and 1GB card widens as your resolution increases. So at lower res the difference isn't that much. At higher res the difference is greater.
The resolution of your monitor will make the difference. Tell us the native resolution.
If it is below 1920x1080, 512mb will probably be just fine.
If it is at or above 1920x1080, then you may benefit from the extra memory of the 1gb model.
For these cards, the only difference between the models will be the amount of Vram.
if he has a 19" 16:9 monitor then his resolution is 1360×768 (or very close to).
and to answer the OP question; you would see little difference between the 1GB and 512mb version of the ATI 5770 at that resolution. Although if you plan to upgrade your monitor to a bigger one, I would go with the 1GB version.