Of course, the OP makes me think that there's not any point in posting here, but oh well...
Both have their solid points. People have mentioned that Americans are generally better at drama, but I only really watch American drama/action/scifi and don't have much access to Brit representatives of the genre so I have no opinion. Obviously there are a lot of really good American shows in those categories.
Comedy-wise, I would say we're even.
Fawlty Towers,
Monty Python's Flying Circus (times ten!),
Yes, Minister,
IT Crowd,
Red Dwarf,
The Office,
Blackadder,
Spaced to a degree... all of those are unspeakably fantastic, and I'm positive there's a few more contemporary ones and others that I simply haven't seen yet.
But America does have
Arrested Development,
Psych,
South Park,
Friends (maybe a controversial one, but I think I've established that I at least know what good TV comedy looks like, so this one's provisional

),
It's Always Sunny in Philadelphia,
Parks and Rec,
30 Rock,
Seinfeld,
Scrubs (at least a few seasons, at least),
Futurama...
The Office! Seriously though, both versions are good, hehe. Anyway, there's a ton more that I just can't think of at the moment.
Mostly I think the perceived differences in quality are part cultural, part access (good American TV, as shown by my list above, does not always equal widely distributed American TV, whereas your sources for Brit TV are comparatively limited), and part matters of national pride. In any one case above, besides a few American shows, I would prefer the Britcom, but nevertheless the sheer quantity of good American programming over the years indicates that American TV does know what it's on about. The argument could fall either way, I guess, but it at least is not an easy decision.