I will probably be gaming on a nice Asus 24" monitor or something like that. Will the 2gb Videocard be better than the 1gb?
The resolution of that Asus you linked is 1920x1080. The general consensus these days is that 2GB is helpful only when gaming above 1080, like 2560x1600. This is only typical on monitors that are 27"+. It won't hurt you to have 2GB, but there's very likely no circumstances that you'll break 1GB at 1080.