Both had their borders drawn by European imperial powers for their own purposes without regard to how it would effect history after the colonies became independent nations. Of course, nineteenth century European imperial powers thought "their" colonies would remain "theirs" forever.
Burundi became independent in 1962 and was at the time of independence a constitutional monarchy under King Mwambutsa IV of the Ganwa people, who claim to be an independent ethnic group but appear to be related to the Tutsi. The Tutsi are greatly distrusted by the Hutus, who are mutually distrusted by the Hutus, in spite of being related as Bantu peoples. Although the region was ruled by a Ganwa king for centuries, the Hutu make up about 85% of the population of Burundi and Rwanda.
The ethnic tensions between the two tribes led to instability in the newly-independent Burundi. The King would appoint a Prime Minister from one tribe and then the other to keep the overall population happy. This resulted int the assassinations of three Prime Ministers and, by 1965, the situation degenerated into a genocidal civil war. In 1966 the King was overthrown by a Tutsi army captain named Michel Micombero. Micombero declared Burundi a republic, named himself president and promoted himself to Lt. General. Micombero used brutality to suppress Hutu opposition, but violence continued. Micombero was deposed by a military coup in 1976 by Colonel Jean-Baptiste Bagaza who was deposed in 1987 in a military coup by Major Pierre Buyoya who gave way to an elected government in 1996 but was re-installed by another coup in 1996 and again gave way to an elected government in 2005.
President Pierre Nkurunziza was elected in 2005, not by a popular election but by an act of parliament. He was re-eelcted in 2010 in what passed for a popular election with 91% of the vote, but the polls were heavily boycotted by opposition parties. He was elected to a third term less than two weeks ago, with the polls again heavily boycotted and only about 30% of voters participating. There was a coup attempt in May that failed.