There are no 'British' or 'American' languages. Both countries speak the same language - English.
I think what you're asking for is differences in AmE and BrE. Specifically what differences are you wanting to know about?
The actual differences are fairly negligible. Probably the most significant is pronunciation, as well as some differences in spelling. There are some differences in vocabulary, but even those seldom cause significant comprehension errors. Context often clarifies the vocabulary differences as well.
There are also minor differences in grammatical preferences when it comes to things like prepositions and verb tense use, but none of these actually interfere with mutual intelligibility. At worst it might sound "odd" to a speaker of the other variant, but it's nothing to interfere with comprehension.
There are also some other minor differences in written style, such as date formats, use of quotation marks, etc.
The only thing that sometimes causes some confusion is differences in vocabulary. I don't know if there's a comprehensive list of these since they're sort of random, but you need to understand that since probably 95% of the vocabulary is the same; as such, it's rarely a major issue.
I think instead of trying to focus on such a broad topic of AmE and BrE differences overall, you're better off asking about specific grammatical differences as they come up.