There are many differences between American and British English. While both are kinds of English, American English and British English differ in a number of ways. I think the one difference that jumps out at you when you compare the two kinds is in their spelling.
American English adopts a simpler approach to English spelling where the spelling of a word usually follows its pronunciation. For example, in British English the spelling of the term that describes the unborn child in the womb follows its Latin equivalent: we write foetus, while in American English we instead write fetus. You see how American English spelling follows the way in which the term is actually pronounced i.e. FE-TUS, while British English spelling does not follow its pronunciation. Several other examples include color vs colour (we pronounce it as COL-OR without the u), favorite vs favourite (we pronounce it as FAVOR-ITE without pronouncing the u) and organize vs organise (we pronounce it as ORGAN-IZE with the z and not the s).
Another example concerns grammar. The present perfect of the verb ''get'' in American English is ''gotten'' while in British English it remains like the past tense i.e. ''got''. Vocabulary concerns another. Famously, British English calls football what Americans call soccer. Americans call diapers what English people call nappies. There are so many other differences. Nevertheless, I think our English language is definitely enriched by these contrasts. In regard to which English, you should study of course I should say both, but I have a sneaky preference for British English since I am of British descent!