It depends on the card. For example, a Radoen 7770 2GB versus a Radeon 7770 1GB with the same memory bandwidth and GPU frequency will have pretty much identical performance no matter what you do except in some situations with Crossfire.
However, a much more high end Radeon 7850 1GB versus the 7850 2GB has a few situations where the 2GB model has the upper hand. For example, BF3 MP at 1080p with some MSAA and maxed out texture quality.
For laptops, there is rarely any difference because the only mobile cards fast enough to make use of more than 1GB in single GPU configurations on the AMD side are the Radeon 79xxM cards. 2GB on anything weaker is mostly just a marketing gimmick (IDK why, but a lot of laptop graphics cards are given like double the memory capacity that they need).