"Moslem" was replaced with "Muslim" for a few reasons (you might also want to review WelfOnTheShelf's answer in the thread on the term "Muhammedism," as the shift from Muhammedan to Moslem to Muslim is linear).

Moslem was initially adopted because it was supposedly the name that adherents of Islam called themselves -- except that the spelling in English produces the wrong sounds. Muslim ("muss-lim") in Arabic is spelled مسلم; Moslem, prounounced "moz-lum," would be rendered something like مازلم -- even if you can't read the Arabic letters, you can tell they're different.

"Moslem" as a term isn't offensive so much as obsolete; since it was intended to replicate the Arabic name and did so poorly, it began to be replaced with a spelling that better approximated the original.