Phonetically, the two words really only vary on the “P” and “B” sounds. They have somewhat related definitions, but they are not at all interchangeable.
Disperse comes with multiple meanings, but they are all largely similar. Basically, to disperse is to drive or send off in many directions. If there is a riot, then the police will work to disperse the crowd. That is, the police will want to scatter the individuals in every which direction so that they no longer form a mob.
Similarly, you can disperse knowledge by distributing pamphlets and other reading materials to the masses. This is to spread the knowledge widely and, once again, in every which direction. In science, they may talk about how a prism can disperse light, meaning to scatter and spread the light.
Disburse, on the other hand, is related to the concept of money and payment. More specifically, to disburse is to pay out, typically for expenses. The company disbursed funds to multiple charities.
It can become confusing, because to disburse can also mean to distribute or to scatter. The equipment manager disbursed the uniforms to the football team. The usage is slightly different that disperse, but you can see how the two terms can easily become confused.
Much like eminent and imminent, disperse and disburse aren’t the most common of terms, especially when it comes to casual speech and casual writing. Even so, it’s important that you know which term is appropriate for which situation and which meaning.
Do you have a suggestion for another Grammar 101 post? I look forward to dispersing more grammar tips to the Internet audience at large.