If you’re using the words to describe a person, then hairy means that the person has a lot of body hair, shaggy means that they have a messy hair style.

Hairy = a lot of hairs Shaggy = long hairs which are quite thick and probably untidy (The exact meaning of course, can vary, depending on the context...)

You have shaggy hair

Note that ‘hairy’ on a person refers to having a lot of body hair, not a lot of hair. A woman with long hair isn’t ‘hairy’. I’m looking shaggy. I better get a haircut. Who does that shaggy dog belong to? (Talking about grooming) Some women like a man with a hairy chest. Some women like a man with chest hair. Some women like a man with hair on his chest. I think ‘hairy’ sounds a bit negative compared to the other two. ‘Hair’ is called ‘fur’ on animals when it is thicker and longer and covers most of the body.

"Hairy" means there is a lot of hair. "Shaggy" means that there is a lot of hair, the hair is of irregular length, and is not neatly trimmed.