I like both of them but I'd like to be more specific:
American: It isn't as nice as British accent but I think is clearer than British accent (at least for spanish native speakers,like me) as It has a stronger pronunciation of the R,a very common letter in Spanish language.The only american accent that I don't like is the one from deep america? I mean Alabama etc...
British:Of course I think the best british accent is the one from the famous triangle London-Oxford-Cambridge,although,obviously, not everyone living in that area has a good british accent.Like Nona said It depends on the background and education of each person.In the other hand,we find Midlands,Northern (england) and Scottish accent.The one spoken in Midlands is not bad (except Brummie accent from Birmigham,It's funny,though).Some accents from Northern england(Geordie,liverpool,Yorkshire) as well as Scottish are a bit difficult to understand,specially if you're not accustomed to them.
We shouldn't forget other accents like Australian,South-African,Irish,specially Irish accent as I think It's rather clear.
Cheers.