Both names mean exactly the same and suffer the same insufficiency as both assign the name of the ruling ethnic group (namely the ‘Bamar’, or ‘Burmans’, or ‘Myanmar’) to the whole country, thus repeating a pattern of discrimination or complacency towards minorities. ‘Burma’ and ‘Myanmar’ are derived from Burmese words (more precisely, from the language of the ‘Bamar’, or ‘Burmans’), which differ only on a stylistic level.
The name ‘Myanmar’ was adopted by the regime in 1989 without any democratic legitimization. While we don’t actually mind using ‘Myanmar’, we would definitely prefer if this change would retrospectively be confirmed by a genuinely democratic decision.
For a detailed discussion, see the page on Wikipedia