If it's a friendship, it's a very strange one. Both have similar goals and similar views in end. Both, it seems, have views of morality that have soured over time into something cynical, but workable. They act as checks on one another (Vimes goes so far as to detain Vetinari during the events of The Truth, albeit for various reasons).
There is respect and admiration there, but if there is any affection (so to speak) in the relationship, it's all on Vetinari's side. I personally think he looks on Vimes like a wayward son, even though, if Night Watch was any guide, the age difference is not that big.