The three significant differences between American and British English are Pronunciation - differences in vowels and consonants, stress, and intonation. Vocabulary - differences in nouns and verbs, especially phrasal verb usage and the names of specific tools or items.
Introduction
You may already know that there is a clear difference in accent. Other differences include some vocabulary and expressions. Less commonly discussed, however, are the variations in grammar. American and British Englishes share almost all of the same grammar.
Differences
There are certain grammatical distinctions between British and American English in addition to variations in spelling and vocabulary. For instance, collective nouns are regarded as singular in American English (e.g. The band is playing). However, the plural form is more frequently used in British English, where collective nouns can be either single or plural (e.g. The band are playing).
British speakers are also more prone to use formal words like "must," whereas Americans prefer to use more casual words like "will" or "should."
The British have long since abandoned the usage of "gotten" as the past tense of "get" in favour of "got," but Americans still use it. In American English, the word "needn't," which is often used in British English, is seldom ever used. Replace it with "don't need to."
The preposition "at" in British English refers to time and location. However, in American English, "on" is used in place of the first and "in" is used in place of the second.
Conclusion
Although there may be some distinctions between British and American English, the two are more similar than they are different. A mistaken substitution of one for the other won't necessarily result in misunderstanding. If you are unable to memorise the subtleties of both languages, don't be too harsh on yourself. Americans and Britons can typically talk with each other without too much difficulty.