My expertise on this exact subject is unfortunately limited by my visiting a field hospital right now. You can ask more questions and I can follow up in detail later.

The short answer is that Burundi had more conflicts sooner, and that Rwanda and Burundi were politically distinct before and after colonialism, and that the states that arose out of them had very different ingredients. The populations are almost identical, but historically and geographically distinct. You had a more established (loosely described as) "kingdom" in Burundi, to the south of the rivers and lakes in the more mountainous South, and it is an area that has proven to be less valuable for mineral extraction than its northerly layer, so it has seen proportionally less interest from foreign powers.

The Burundian state took a more proactive roll in suppressing its majority population, and you can read up on the genocide of 1972, the genocide of 1988, etc. The power elite that took control of Burundi were highly localized between a few villages so you see political alignments more as struggles between members of specific communities than as a Hutu-Tutsi struggle. Violence, even during major genocides is often highly localized depending on the communities - heavily populated cities would see a large concentration of violence, and more stable communities would see less. I think this is a pretty clear commonality that Rwanda and Burundi would have, with the armies of both being closely controlled by Tutsis.

As for the historical facts of the 1994 Genocide, the assassination attempt is largely seen as being targeted at Rwanda with Burundi just coming along with the ride - this is highly debatable but it saw a reassertion of Tutsi power in Burundi against an already embattled community that was unlike the huge outpouring of pent up anger and frustration in Rwanda.

I hope that gives you an intitial feeling of what was going on. A good introductory book on how to think about Burundi would be:

Uvin, Peter. Life after Violence: A People's Story of Burundi. London: Zed Books ;, 2009.

It has problems but is a good starting place.