To disburse is to pay out or to expend funds. Without exception the word relates to money. To disperse is to scatter or to cause to vanish. It is possible for money to be dispersed, but that’s usually a bad thing; money that is paid out or expended for a given purpose is disbursed.
Disburse‘s corresponding noun is disbursement, not disbursal. Disperse‘s corresponding noun is dispersal, not dispersement.
The words are often mixed up—for example
As Greece waits for its euro-region partners to disperse funds, the European Union has announced no concrete plans to help other nations should aid be needed. [BusinessWeek]
Cleanup crews are spraying something similar to dish-washing soap on the oil slick, trying to disburse it. [News 4 Jacksonville (article now offline)]
Following an application process, the local boards decide how to disperse the money to local agencies. [Washington Post]
And these writers use the words well:
Instead of a dark slick on the surface, it becomes tiny droplets that disperse into the water below. [McClatchy]
The Education Department will handle loan applications and disburse money to the school’s financial aid office. [Chicago Tribune]
Biles said guardsmen used rifles fixed with bayonets to disperse the protesters. [Marcos Island Florida (article now offline)]
The fund expects to start lending money in the next couple of months and disburse most of the $200 million in 12 months. [BusinessWeek]