"You're not the boss of me" vs "You're not my boss"

"You're not the boss of me" emphasizes "me," and is something a child would say (or we imagine would say) to his parents. It's also often said by an adult to give the statement that connotation, sometimes in an ironic way.

"You're not my boss" is a bit more on-the-nose and declarative, whereas "you're not the boss of me" is more absolute, confrontational, and draws more upon the cultural shared knowledge of that phrase.

The big spike at just after 2000 (i.e. 2001) most likely comes from the popularity of the TV show Malcolm in the Middle which has the title song Boss of me by They Might Be Giants, with the chorus:

You're not the boss of me now!

You're not the boss of me now!

You're not the boss of me now!

And you're not so big.

The TV show was very popular and has been broadcast all over the world.