Basically, neither is an accurate transcription into phonetic English of the local pronunciation of the name.
Though the words look radically different in Roman scripts, in Burmese they are pronounced almost identically: with a quick, unstressed first syllable, either “buh” or something like “munn”, followed by a longer “MA”. In neither name is there a hard “r” sound anywhere. It is never pronounced “MAI-an-marr”.Similarly, in the mid-1980s a guide in Beijing told us his girlfriend was a local, he said she was "Pekingese." The other DrC and I were proud we managed not to burst out laughing at the mental imagery that term invoked. He'd never have understood our mirth.
Gustaaf Houtman, an anthropologist who specialises in the country, explains that native speakers use both words: Myanmar is the formal, literary form and Burma an everyday term. Burma has the advantages of ease of pronunciation (for foreigners), and visual consistency: the adjectival form is Burmese.
