American English isn’t very different from British English. There are a few differences of grammar and spelling. But since modern British English is heavily influenced by American English, some of the contrasts are disappearing.

Grammar
Here are examples of the most important differences of grammar.

Simple past
In American English, the simple past is often used to give news. In American English, it is also possible to use indefinite past time adverbs like already, yet, ever, just and before with simple past.
in British English, the present perfect tense is used with indefinite time adverbs.

compare:

He just went home. OR He has just gone home. (American English)
He has just gone home. (British English)

Have

In British English, do-forms of have are usually only used to express habit or repetition. In American English, do-forms of have are more common.

Compare:

Do you have a problem? OR Have you got a problem? (American English)
Have you got a problem? (British English)

In British English, have is possible in short questions and answers. In questions and negatives, Americans usually use do-forms of have.

Have you an appointment? (British English)
Do you have an appointment? (British/American English)
She has not the charm of her sister. (British English)
She does not have the charm of her sister. (British/American English)

Got and gotten

In American English, the past participle form of get is gotten. In British English, the past participle form of get is got.

I have never really gotten to know her. (American English)
I have never really got to know her. (British English)

Subjunctive
In British English, subjunctive forms are formal and unusual. British people usually prefer should + infinitive in that-clauses. Subjunctive forms are very common in American English.

Our advice is that the company invest in research and development. (American English)
Our advice is that the company should invest in research and development. (British English)
It is important that she be informed. (American English)
It is important that she should be informed. (British English)