Distribution
Distribution refers to the way that objects are arranged or organized. Another way to look at this is to measure the normal collection of places or times a certain event takes place or a particular object is located. The key to understanding this is the indigenous or at least planned sense that "distribution" connotes.
Example: Plant Life
When you talk about a plant's normal distribution, you're talking about the habitats where you would normally find it, and the concentration at which that plant tends to grow. For example, you would expect plentiful distribution of palm trees in Florida, but not so much in Wyoming.
Dispersal
Dispersal refers to the spreading out of people, animals, seeds or other items to different places. It has much less to do with an orderly plan or set of indigenous conditions and much more of a spontaneous feel to it. When riots are about to begin, for example, police officers will generally ask for the crowd to "disperse" -- to spread and go somewhere else.
Example: Plant Life
Plants disperse seeds within their distribution area, but each plant has a fixed dispersal range. In other words, their seeds can only carry so far in the wind. Plants that have a relationship with birds or bees to pollinate for them have a larger dispersal zone than those that do not, but every dispersal range has its limits.