To see the overall performance from your new graphics card can't be measured from memory size alone. It also depends to the game you play, total display, and most importantly, the resolution.
The same case applies to a computer with 2gb ram versus 8gb ram. For a computer with the same processor you will see the contrast performance between two case above. As you can see, a computer with more memory can do more task.
The video card with bigger memory has more advantage since it can manage more task hence it will have better function if you play on 1080p resolution or on dual monitor. But if you are still using one monitor with lower resolution then 1gb graphics card is sufficient.
GTX 750 ti 1gb for example, even though can run games smoothly on 720p and 900p, it will struggle on higher resolution due to limitation on the memory size.
As the technical reason, the heavier task a computer must run a games, the more vram requires. Hence, don't get surprised when your video card is struggling to run game on larger resolution.
In the end when you have to get Graphics Card with larger memory?, well when you want to play games and activate additional graphics feature like anti aliasing, physx, blur and other fun features. Moreover, if you want to run games on 1080p then get at least 2gb size graphics card.
For the usage of dual or three monitor as mentioned above, this graphics card type is more reliable.
And remember, there are bad optimized games out there like Grand Theft Auto IV where it needs more resources then most of modern games. Moreover, you can find a lot of missing textures here and there while playing the games with 1gb Vram.
Of course, it's always recommended should you have more budget, to get faster and video card with bigger vram since the AA games will demand more power and resource from your computer everytime it releases. However, if you are still living with a 720p or 900p old monitor in your room then you won't really need a 2gb memory type video card.
