Cons: Cost, weight, could damage knives
Pros: Cheaper, lighter
Cons: Wears down quicker, could trap bacteria and spread to other foods
The main difference in the two is that wood is porous so capillary action causes the microbes to be pulled into the wood, away from the surface and the bacteria gradually die inside rather than multiply. When you cut plastic boards, you may notice that after a while there will be slight grooves that form as you wear down the board and cut into the plastic. These areas aren't always thoroughly cleaned and bacteria are able to live and multiply there, especially in the microscopic cuts. This is especially a problem if you use the same cutting board for meat and vegetables where cross contamination could happen. What some people do is designate one side of the board for meat and the other for vegetables and flip it over.
You can clean wooden board and then microwave it to kill everything. Don't over heat it or it could damage it. Plastic boards can withstand cleaning chemicals better than wood so you can bleach these boards to kill everything.
Depending on the board and how hard it is compared to the knife blade, it could dull the knife over time. Very hard woods and glass boards could do this so be careful. Remember it has a finite life. If the plastic is badly damaged, throw it out.
Otherwise, they both get the job done. Just make sure you clean it thoroughly...... Unless you like salmonella