Wow, I've never had to think about the difference until now. I'll try my best to explain.
Friendship specifically refers to a bond between two or more friends.
ex)
Our friendship is great. - we are good friends/ we have a good connection, we've known each other for many years and we are very close.
Relationship doesn't necessarily imply anything romantic. It's more flexible and can refer to any kind of bond shared between two(or more) people. Whether it be romantic, among friends, family, one's teachers/co-workers, soccer coach, etc.
For example I can say:
'My relationship with my father is non-existent" meaning we don't have a bond at all, we are not in contact and I am not close to him.
"My friends and I have a good relationship" my friends and I are very close and we have a good bond. We are there for each other and we also help each other with any problems one another has.
"His relationship with his girlfriend is very bad." They argue very frequently, they don't see each other often and they don't talk much and they are on the verge of separating.