Although generational improvements are usually small, AMD had been somewhat "behind the curve" with their HD 3850 / 3870 and 3870X2, as well as their HD 4850, 4870, and 4870X2, so their 5000s and 6000s did receive healthy generational improvements. Normally, between a 3870 and a 4870, the feature set improves, while overall performance barely goes up much, less than 10% or thereabouts. Normally, then, an HD 4850 might not be as powerful as a 3870 had been.
The very first number is the generation. The second number is the CLASS, and 800s are top of their line. The third number is differentiation within class. I think if we compared a 4870 with a 5850, the generation improvement would, or should, absorb the CLASS difference. Anyway, let's use GPU Boss to get a feel for your 6800 (there were at least two in the class, probably three, you may use any number of GPU inventory softwares to learn which you have.
Last I heard, GPU-z was free.
According to GPU Boss, your card HAS to be better, unless there is an oddball "6830" in that PC you have (there *was* one of those for the HD 4800s).
http://gpuboss.com/gpus/Radeon-HD-6850-vs-Radeon-HD-4870
The HD 6870 is better still, of course.
http://gpuboss.com/gpus/Radeon-HD-6870-vs-Radeon-HD-6850
Laptops cannot actually be compared to their desktop cousins, because the chip producers have less ability to enforce any standards. There is an HD 6830, an "m" for Mobile, and it's comparatively terrible:
http://gpuboss.com/gpus/Radeon-HD-6850-vs-Radeon-HD-6830M