American English and English English (also called British English) grammar are considered identical. English English is most often taught to students around the world (although some countries are now beginning to embrace American English instruction). There are dialectical differences, like the use of have and got and the accepted collection of fixed phrases that do not require articles or possessives (e.g., UK: at University; US: at a/his/the/my etc university), but these do not comprise grammatical differences. There are some punctuation differences though of late American English punctuation is gradually coming more in line with English English punctuation, pertaining especially to use of commas.
Grammar is defined as a formal (having form) description (for descriptivists) or prescription (for prescriptivists) of language and it's rules governing morphology and syntax in the verbal and/or written formation of words, phrases, and sentences. There are many dialectical variations in the wide number of varieties and dialects of English, but rules governing the morphological word formation and syntax of Standard American English and Standard English English are the same.
Both AE and EE follow the Subject Verb Object or Subject Verb Object Adverbial (SVO or SVOA) grammatical rule. There are of course variations to the SVO and SVOA grammatical order, but they are just that--variations to the rule. Another way to state the grammar common to AE and EE is to say that they both, by grammatical rule, follow the someone did something to someone/something somewhere at some time for some reason and/or for some purpose model.
The above model substitutes the SVO/SVOA prescribed grammatical definition this way: someone/S did something/V to someone/something/O somewhere at some time for some reason and/or for some purpose/A, yielding SVO or SVOA. Yet another way to demonstrate the grammar common to both AE and EE is to ask who did what to whom where when how and why. This wh-question model substitutes SVO/SVOA in this way: who/S did what/V to whom (or what)/O where when how and/or why/A. Ginger Rogers indirectly references this wh-question model when in Follow the Fleet she asks Hillary Hilliard, "Who's leaving what where when?"
[Let's analyze my last sentence into the SVO/SVOA pattern: Ginger Rogers/S indirectly references/V this wh-question model/O when in Follow the Fleet she asks Hillary Hilliard, "Who's leaving what where when?"/A. You'll notice that I've inserted the additional adverbial phrase "in Follow the Fleet" into the when-clause filling the A slot. It could also have been written as this: Ginger Rogers indirectly references this wh-question model in Follow the Fleet when she asks Hillary Hilliard, "Who's leaving what where when?" Adverbial phrases and clauses are the element of AE and EE grammar most open to variations in placement.]
I could find that there is a great many differences between American forms of the language and the British version of it. In my mind, pronunciation and diction seem to be the largest difference. I would say that there is much more geographic diversity in America in relation to language. This is present in England to a great extent, but in America the different regions' pronunciation of the language helps to bring out a different nature to it. For example, the dialect in the Southern part of the United States holds much difference than the far North Eastern section. In this light, one can see an almost "new" version of the language emerge.