It depends on what you want out of chickens. I'll give my opinions.
Pros
1. Hybrid vigor - Increased vigor or other superior qualities arising from the crossbreeding of genetically different plants or animals. Also called heterosis. It is documented that this does occur, but "superior qualities" may mean different things to different people.
2. You can select to breed the ones that display the qualities you want without worrying about breed. Your goals may be different, but I'm breeding for better meat birds with a lesser emphasis on egg laying. I am also selecting against behavioral traits I don't want, like birds that like to fly out of the run or a laying hen that resists a rooster's advances. An uncooperative hen disturbs the peacefullness of my flock and hurts the fertility of my laying/hatching flock.
3. You can get some really interesting colors and patterns after a couple of generations. This also makes it easier to tell the chickens apart.
Cons
1. If you are going to sell the eggs as hatching eggs or sell the chicks, it is easier to sell purebred. If the eggs are for eating, it does not matter.
2. If you are going to show the birds, they need to be purebred.
3. If you maintain the purity of your breeding stock, you may be able to get sex link chicks. I think a buff orp male with barred rock female will give you sex linked chicks. This is lost after the first generation.
4. If you have two separate breeds and want to maintain the purity of the breeds, you have to keep them separated by breed. This takes more management and more expensive facilities.