The simple answer is this - not very much. There are actually very few differences between the speech of a user of standard American English and that of a user of standard British English. Yes, some vowels are different - notably the 'o' and the 'a' - but not all vowels are different. The 'e' and the 'i', for example, are often virtually the same. The other main difference is the pronunciation of the 'r' after a vowel. Most speakers of American English will pronounce the 'r' at the end of 'car', while most speakers of British English will not. But note that this doesn't apply to all speakers - in many areas of the British Isles (most significantly Scotland, Ireland and SW England) the post-vocalic 'r' is pronounced, while a minority of US English speakers (such as those with the famous 'Harvard' accent) don't pronounce it.
There are also a handful of words which have a different stress pattern eg ballet.
Here's a fact. If the average Chinese person listens to a conversation between two educated speakers of English who have neutral accents, one American and one British - say, an academic from Oxford and his/her counterpart from MIT - they probably would have to listen hard to tell which was which. Some vowels and the 'r' sound would be different, but otherwise the two speakers would sound pretty well the same to the non-native ear.
Many foreign learners imagine that American English and British English are two entirely different things, like Mandarin and Cantonese for example. This is not the case at all. In fact, the differences WITHIN the two countries are greater than the differences between them - there are enormous variations between the speech patterns of different regions within the UK, but comparatively few between the standard speech of, say, a British RP speaker and a speaker of general American English.