What Thorsen said.
Quite simply, games (and everything else) will look sharper and much better.
If you are planning on buying a higher end Graphics card, you really will want a higher res monitor.
Although you didnt say which card you were looking at, lets assume its a GTX670/HD7950?
It will be harder for the GPU to render the images at 1920@1080vs 1368X768. (More Pixels to draw)
Which is why if you want to run your games at Max settings, 1080P, you really want a higher end card such as these.
However, its worth mentioning that at 1368X768, it takes far less power to render at those same settings.
So a higher end card may be a waste of money and power at that resolution if there are no plans to upgrade to high res monitor. You can get by with cheaper and get the same performance. At least at the current resolution. If you do upgrade the moniotor, the lesser card will likely struggle at the higher resolution, where the higher end card will just start to stretch its legs.