English speakers seemingly use the word Burma or Myanmar to describe that country. My impression is that it is somewhat interchangeable. If you use Burma instead of “Myanmar” it is some how ok—you just sound a bit old-fashioned, which is perhaps how the United States Embassy in “Burma” sounds to ears inside Myanmar. On the other hand, some who are in opposition to the current Myanmar government prefer the more traditional name of “Burma,” and favor it when speaking English.
Burma of course is an older name for the country, but only sort of. Myanmar as a word has been applied internationally to that country only since 1988, when the long-term dictatorship of General Ne Win was overthrown by another group of generals. The generals who began the oppressive “SLORC” regime reached back into Burmese history, and adopted the word Myanmar as the name of a country. Myanmar was a word long used in literary circles to refer to the country. The SLORC preferred this they said, because the term Burma was associated with British colonialism. They also changed the name of the capital from Rangoon to Yangon to reflect local pronunciation, and adjusted the English spelling of a number of other places to conform to new transliterations, and as an expression of political authority.
The term Burma itself is an adaptation of the older term “Bamar” which is an ethnonym referring to the people of the central plains. This word in the English language became “Burma,” and in the Thai language roughly “Pama,” which is still commonly used (though so is “Myanmar”). The colony the British ruled from from the 1800s to 1948 was referred to as “British Burma.” At the time of independence in 1948, the new country became known simply as Burma, which had the Burmese language as its official language, and a citizen of the nation (and the largest ethnic group) was known as Burmese. But as a national ethnonym, this did not sit well as a global term for the dozens of groups which went into revolt after independence for any variety of reasons. But the name of the country remained straightforward.
The asserting of the new name “Myanmar” in 1988 by the new military government was not always widely acknowledged. Groups in revolt continued to use the word Burma and Burmese to describe the government they were in revolt against. The US government also did not recognize the legitimacy of the new name, and continues today to refer to the country as Burma, while its ally the United Kingdom refers to Myanmar. Indeed the US Embassy continues to use the word Burma on its official website, a situation not obviously objected to or blocked by the Myanmar government.
But for the majority of people living in Myanmar today, the use of the word Myanmar is not controversial. And they refer to themselves as the “Myanmar people” who speak the “Myanmar language.” If you use the word Burma when speaking English, it is just considered a bit archaic, but hardly controversial.
As for the rebel groups, and some of the human rights groups interested in the country, the term Burma is often common and preferred. However at least the people I’ve talked to from such groups, using the word Myanmar is non-controversial.
For me the lesson is that when I speak English, I continue using both, sometimes in the same conversation. Neither word is considered derogatory. Rather I try to be sensitive to the context of whom I am speaking with, and what I am speaking about.
If want to think of analogies, some examples might include Siam/Thailand, Helvetia/Switzerland, Cambodia/Kampuchea, Tanzania/Tanganyika, Russia/Soviet Union, United Kingdom/Britain, United States/America, etc. etc. Indeed I suspect the list could be quite long!

Tony Waters is czar and editor of Ethnography.com. He came to us from the Sociology department at California State University at Chico where he has been a professor since 1996. In 2016 though he suddenly found himself with a new gig at Payap University in northern Thailand where he is on the faculty of the Peace Studies Department. He has also been a guest professor in Germany, and Tanzania. In the past, his main interests have been international development and refugees in Thailand, Tanzania, and California. This reflects a former career in the Peace Corps (Thailand), and refugee camps (Thailand and Tanzania). His books include: Crime and Immigrant Youth (1999), Bureaucratizing the Good Samaritan (2001), The Persistence of Subsistence Agriculture: Life Beneath of the Marketplace (2007), When Killing is a Crime (2007), and Schooling, Bureaucracy, and Childhood: Bureaucratizing the Child (2012). His hobby is trying to learn strange languages–and the mistakes that that implies. Tony is a prolific academic, you can read more of his work at academia.edu.or purchase one (or more!) of his books from Amazon.com.