Are tags different than categories?

This is indeed strange behaviour. The slug is a unique key in the table that stores taxonomy terms, if you make sure to explicitly set the slug to something different when you create the term, they will not be linked. However, once you’ve created a tag and a category with the same slug, it appears they are forever connected, unless you delete one and recreate it with a new slug. I don’t think this was always the case though, I distinctly recall WordPress appending -2 to the slug when attempting to create a term that is the same as a term in another taxonomy.