The gramatical differences between "high" forms of British and American English are comparatively small. Certainly the dialectal variations within each country are further departures than standard American English is from standard British English. In each the differences are more of preference and still usually considered correct in the other.
A more apt example would be Swiss and high German, which are (aside from the fact that Swiss Germans are also fluent in high German) mutually incomprehensible. To a high German speaker, Swiss German sounds decidedly silly and has moderately different grammar. However since there's no conflation of social class with the Swiss German dialect, the comparison of Swiss and high German is much less politicized than AAVE and standard American English.
Austrian German, on the other hand, is virtually identical to standard high German grammatically – it just sounds funny and has a few regionalisms in its vocabulary: kind of like the difference between American and British English.
I heard recently that Arnie offered to do the voice for the Terminator in the German dub of the film of the same name. They said no because his accent made the killer robot sound like a farmer or country bumpkin to German ears.
Honestly, his accent in German is hilarious. I dug up some old YouTube footage of him in German a while back and he sounds ... cute. Definitely not hard-core like he does in English. ;-)
Linguistically, there's no agreed-upon rule for what makes a "dialect"[1]. What I heard from Linguistics people is that they're all just separate languages, some with and some without a written form.
My usual half-serious stance on this is that a language is just a dialect with an army. But this does of course break down in the case of Icelandic, which is definitely a language, but has neither, but clearly captures tha fact that answers to these questions are to a large part political. When I was younger, there was a language called serbocroatian, spoken in one country and written with two different scripts...
A more apt example would be Swiss and high German, which are (aside from the fact that Swiss Germans are also fluent in high German) mutually incomprehensible. To a high German speaker, Swiss German sounds decidedly silly and has moderately different grammar. However since there's no conflation of social class with the Swiss German dialect, the comparison of Swiss and high German is much less politicized than AAVE and standard American English.
Austrian German, on the other hand, is virtually identical to standard high German grammatically – it just sounds funny and has a few regionalisms in its vocabulary: kind of like the difference between American and British English.