What are the basic differences between American English and British English in terms of grammar?

Grammar is generally the same, but there are cases of different verb usage and /or various spelling - 

I have not got

I do not have

colour

color

as far as Im aware - basic grammar should apply in both e.g. the verb and the subject must agree in number and gender, etc. However in American English they tend to spell words the way it is pronounced, whereas in British English spelling and pronunciation is influenced by the ancient languages of the celts, welsh etc.  

I agree with Mr. Popov's answer

Differences are at the level of spelling, vocabulary , pronunciation and grammar