More VRAM generally allows for higher resolutions, more detailed textures etc.
But you still need the GPU power to actually drive those higher resolutions or graphics settings. So for a given GPU, there's only so much memory you're realistically going to benefit from; beyond that, your GPU will just hold you back regardless of how much memory you have.
In the case of the 7770, 1GB is almost always going to be enough. There may be a few special situations where 2GB would be an advantage, but it would really be rare.