The main reasons why graphics cards come with more memory.
Relate to how powerful the model of the Gpu is on the card.
This allows you to not only display higher screen resolutions providing you have a monitor to display the resolutions.
But also the higher the resolution the more memory of the card is required to display the higher resolution.
Also it means you can crank the graphics quality of games up and the amount or FX the card can produce.
When you set a graphics card display resolution higher. and in game graphics and effects it requires more memory of the graphics card. as higher resolution texture require more memory space.
It can also help with in game loading where the actual textures are stored on the graphics cards memory for faster access. If the card has less memory on board from time to time you can experience a slight glitch or lag.
Because the card has to call textures for example from system memory to the texture memory of the graphics card.
Experienced as a slight split second lag while new textures are loaded for in game texture mapping.
But for system with a 1080p resolution most cards perform very well even with 2Gb of card memory over a 4gb enabled model.