By functional organization, I'm assuming you mean a hierarchy. For example, the Engineering Department of a company I worked for had a VP. Section managers reported to the VP (Operating System, Applications, Communications, etc.). Subsection managers reported to the section managers (Communications was divided up by the 4 different network devices which were supported). Unit managers reported to the subsection managers. One-over-one management was practiced since it would have been very inefficient for all decisions to go up and down the entire chain of command.
Matrix management continues to have a functional components such as marketing, sales, engineering, and operations. There are product components that also have managers that cut across these organizations to produce a product. The benefit of this arrangement is that there is knowledge sharing across product lines within a discipline (e.g. all the engineers share information). It also allows for people to specialize in a discipline and have more professional opportunities. Project managers are responsible for delivering the product on time and within budget.
The disadvantages of the matrix organization are that there are often more managers required (functional and product), which increases general overhead costs. In a similar manner, decision making takes longer because more managers need to be involved. Conflicts of loyalty to product vs. functional managers can develop.