Writer is someone who writes articles, books, e-books, etc.
Blogger writes opinions or articles in a blog format.
There is a difference, not so much as being published, but if you are paid. There are lots of bloggers who write on their own blogs. Their writing however may not qualify them to write for companies who need freelance writers and bloggers.
Writers however can both write for money and for fun. The standard of writing held higher. Word usage, sentence structure, grammar at a different level than the casual blogger.
Now ask me what Copy is and that will start a whole different discussion.