I've seen this question asked a few times by Arabic speakers.
Rest assured that the differences between British English and American English are nowhere near as many as the differences between the different Arabic dialects. There are probably more differences between the dialects of Gulf Arabic alone than between British English and American English.The differences between British English and American English are mostly in the accents. Arabic speakers often don't know the difference between an accent and a dialect. Accents are the ways people pronounce things. There are some differences in grammar and vocabulary, but they're minimal, and pretty much everyone can understand everyone else.