That's a very difficult question to answer, because there are many different accents in both British and American English. Someone from Boston sounds very different from someone from the immigrant Italian community of New York, and they both sound very different from a Texan farmer.
Generally, though, there are differences in vowel sounds, particularly the short vowels. British speakers often don't pronounce the R in words like "car" and "hard" but Americans nearly always do.
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