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Burma or Myanmar? What's the difference? Does it matter? A thread on political linguistics in Southeast Asia 
First, what's the difference between "Burma" and "Myanmar"? You might be surprised that these are basically the same word, borrowed into English twice from two different registers of the Burmese language. (A "register" describes the level of formality of a language)
Burma comes from "Bama." Myanmar comes from "Myanma." "Bama" (pronounced buh-mah) is a colloquial version of "Myanma", which incidentally has two syllables (myuh-ma), not three
How do you get "b" from "my"? Well, make a "b" sound and notice where your lips are. Then do the same with an "m" sound. Both of these are voiced bilabial consonants, and such sound changes are pretty common
For a rough analogue of Myanma/Bama, say the name of the country that is the six counties in the northern part of Ireland. Is it called "Northern Ireland"? Or is it "Norn Iron"?
So moving on: the colloquial "Bama" is the root of our word Burma. It was borrowed into English during colonial times, if not before. It is also the name of the majority ethnic group, the Bama, which are properly called Burman in English
And here is where politics comes in. Burma is a multiethnic state with a number of non-Burman ethnic group (Shan, Karen, Mon, and dozens of others). What should the word for the country that encompasses all of these peoples be?
One idea is to use Myanma. And recall, this is also the way to say "Bama" in the high-prestige register of the Burmese language. So it feels both more formal, and more separate from the Bama people (but let's note that etymologically, it's really not...)
What about that -r at the end of Myanmar? Well that's not a sound when using Latin orthography to write the Burmese language. That -r actually marks a tone. So you do not say it. Say "myuhn-ma" instead. (The "uh" is a schwa, written É™, like the second syllable in "pencil")
OK OK moving on for real now. Now it gets *really* political. The decision to switch from Burma to Myanmar in English happened in a period of great political turmoil in Burma, following the 8888 uprising https://en.wikipedia.org/wiki/8888_Uprising
The use of Myanmar is associated with the military junta that ruled the country in one from or another from 1990 until the liberalization in the 2010s. (Analogously, Cambodia was termed Kampuchea under the Khmer Rouge. Same story etymologically.)
For a long time, many people--I among them--used the word Burma instead of Myanmar when speaking in English as a way to signal opposition to the junta-led government.
Since the return of Aung San Suu Kyi to formal politics, this convention has eroded. Using Myanmar instead of Burma became rather less politically loaded. People like me became inconsistent in using one versus the other.
Of course, even if you were to use Myanmar exclusively, you still have to use "Burmese" as the adjectival form. There is no "Myanmarese". And if you want to talk about the country's plurality ethnic group, you need to say "Burman." So the ambiguity is not going away.
Some people have noted that the US State Department used Burma not Myanmar in its most recent press release about the ongoing coup https://www.state.gov/burmese-military-arrests-of-civilian-government-leaders/
But this isn't new. The US has always used the name Burma officially. https://www.state.gov/u-s-relations-with-burma/
So now you know the tricky history of Burma/Myanmar/Bama/Myanma/Burmese/Burman. The terms differ because of linguistics and colonialism, and they matter because of politics.
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