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no it does not necessarily improve the clocking prowess of the GPU but it helps when handling larger screen resolutions.
Generally 1gb is ideal for anything upto 1920 x 1080 and 2gb for anything beyond that resolution.
eg a 1gb card may struggle with a monitor capable of 2560x1440 but a 2gb card would be fine.
Are there apps available to monitor how much vram is being used during a game?
Lots of game siv played recently have all used more than 1gb or ram in one way or the other.
DA2 and presently shogun 2.
DA2, needs more than 1Gb to use the high res texture pack.
Shogun 2, well putting the settings to a high enough IQ, uses more than 1Gb of ram, whether or not it needs to at a hardware level, no idea, but the game devs made it that way, so as it stands the game can use more than 1gb of ram if you crank up the settings.