Historically, arse was reserved for the rump and ass for the domesticated ungulate now called donkey (itself originally a slang or dialectical variant). However, ass has long superseded arse in both American and British English as the preferred spelling for both rump and donkey. Arse remains common in British English slang and in the writings of those who cannot bear to adopt linguistic changes popularized in American English.
The phonological loss of the r is not uncommon (e.g., curse to cuss, baers—the fish—to bass). In this case, the shift from arse to ass is attributed to nautical slang.