It is much like the situation Lee had with Neji, really. Neji is a true genius, the process of learning and training is fast and easy for him. Remember Neji's thoughts about geniuses and that they are unreachable bu people without talent?
Lee is a perfect example of hard-worker. He had absolutely no talent neither for ninjutsu nor for genjutsu. He trained extremely hard for years, and his willpower and stubbornness (and Guy-sensei) helped him to finally become what he was.
Sasuke, much like many others inUchiha clan, was a genius too. While in their first encounter his speed wasn't enough to match Lee's, in the second one, he was already able to reach the same speed. Of course, the Sharingan also helped. No, it can not "copy the speed", I think it's portrayed nicely in the anime (though I dont remember the episode), when Sasuke can see Lee's movements with Sharingan, but cannot react fast enough with his body. However, Sharingan also helps in practicing Taijutsu, because it lets it's owner learn techniques faster. Of course, it helps in battle as well, allowing to see enemy movements absolutely clear and precise, and possible copying them (Sasuke showed that in the first part on exam, when he used Sharingan to copy the answers to the test from the guy in front of him by copying his arm movements).