4 GB graphic card has more Graphic RAM compared to a 2 GB graphic card. One with a larger chunk of memory will be better. A normal desktop or laptop monitor would be good enough for a 1 GB graphic card.
If you are going to run games on a 1080p screen, you probably need a better one with more Graphic RAM.
2 GB+ graphic cards are used by people with multiple-monitor setup, Hardcore gamers and professionals. That is to plug in more than one high-res monitors, which , requires more amount of graphic memory.
The performance of your card depends upon the card's generation,its clock speed and its GPU capacity and family.