it basically has to do with the amount of strain it takes off of system memory, at least from my understanding. Dedicated graphics memory is faster for the graphics card to access, and also keeps graphics assets off the RAM the rest of the system uses.
It doesn't keep anything of the system memory. System memory still has to store loads of stuff. The graphics memory usually cannot hold enough and the in some more streaming games like Call of Duty, MS Flight Sim. the Memory is constantly swapped at some GB/s. Which means the entire content is replaced many times each second. You need to have that in RAM just to swap back and forth fast enough.
As soon as you have to little VRAM textures need to be swapped. And they suddenly pop up during you run around in a level. In GTA 4 that already happens if you 1GB RAM, a game that gladly takes 2GB.
If you run out of VRAM that is not only less bandwidth it means a latency penalty and that really kills performance. Too little VRAM limits the detail settings that you can set. It is like with system ram, as long as you have enough a little more or less makes no difference. When you have to little it means more swapping and that hurts performance. Yet unlike system memory it also limits the stuff you can do at all, as some games just won't let you enable settings if you don't have enough VRAM. You get different options with too little VRAM.
Some settings like high detail textures are no problem for a GPU speedwise but they give quality but with too little VRAM you cannot fit them in there. Enabling AA increases VRAM use and if that pushes over the 512MB you are in for a huge decrease in performance that would otherwise be not even half that.
More VRAM means more graphical quality. With the right setting you can still run 512MB but it is already at the very edge on many modern games. The 650M can push a lot of pixels and 512MB is just not enough.
The 560M is equally fast and was sold last winter with 1.5 or 3GB.
A 650M should come with 1GB minimum and 2GB would make it future proof.
They should have put 1GB GDDR5 in the lower end and 2GB into the highend.
512MB is barely enough today and will suck 1-2 years from now. Considering the speed of a 650M it already sucks. 2GB cheap DDR3 would have been better. Which means you can find a faster GPU in some notebooks that cost less than half that or in some thinner ultrabooks with 15".
512MB is a bad joke. If they wanted the highend to look better they should have made the difference between 1 and 2 GB.