It is a big difference. Text on an 1920 x 1080 15.6" screen will look noticeably smaller than 1366 x 768 on a 15.6" screen. I recommend you go to a computer store and check it out yourself. However, a computer may not have a laptop with 1920 x 1080 resolution.
HD movies will look better on a 1920 x 1080 because that is the resolution HD movies are in. Well... almost... Movies in 16:9 aspect ratio will be 1920 x 1080. Movies shot in 2.35:1 aspect ratio will be 1920 x 816 which means you'll have black bars on top and bottom. Shrinking a HD movie down to 1366 x 768 will result in less sharp video, but you have to be pretty close to be able to note the difference in most cases. On such a small screen it shouldn't be much of an issue.
Gaming wise, there will be a huge difference in performance. The higher the resolution the lower the performance. So if you want to play games at 1920 x 1080, then you better get a powerful video card unless you don't mind lowering the resolution. For a 1920 x 1080 screen I would personally go for a Mobility HD 5850 (I don't know the nVidia equivalent) which is roughly the equivalent of the desktop HD 5770 in terms of performance. Therefore, not only will you need to spend more money on a higher resolution screen, you also need to spend money on a more powerful video card.