The output resolution doesn't make much difference once you get to this sort of memory scale. 1920x1080 at 32 bits per pixel is ~8Mb, so even if an app/game/what-ever is triple-buffering the memory used by the frame buffers will only be ~24Mb.
The extra memory will be used for thing like higher resolution textures (improving some visuals) or just being able to hold more texture data on the card saving trips to and from main memory (improving speed when moving between areas that might need different textures) and other such. How much difference this will make depends on the apps/games you plan to run - some will be able to take advantage of the extra while many will ignore it. Another related issue is that if the game/app does user more RAM for higher resolution textures the GPU will have more work to do presenting those textures on screen, reducing framerates so you might tone the settings back down anyway. Many game/card reviews compare benchmarks between different RAM sizes as well as different GPUs, so I suggest you try find some such resources (try "gts250 1Gb 512Mb benchmark" in a search engine as a starter-for-ten) to get an impression of what difference to expect and if that difference is enough for you to care about.
If you look at most game specs "256Mb minimum, 512Mb recommended" seem usual these days, so my gut suggests that unless you are a big gamer type, or the price difference is quite small, I would stick to 512Mb as you are likely to want an upgraded GPU before the amount of RAM becomes a major limitation. If the price difference is large you might be better of spending the extra on a game to play with the new card instead! But check out some relevant reviews and benchmarks to be more sure.,