As a native speaker of Tagalog, I don't see any difference between Tagalog and so-called "Filipino". IMO, the name "Filipino" is used to easily identify it as a language from the Philippines. Since Tagalog doesn't really indicate it. But as far as I know, I've never heard anyone say, "I speak Filipino."
It's way too vague to describe a language of country with more than 90 languages (not sure how many) as simply "Filipino". It's almost like a person from for example, Vancouver saying "I'm from earth"
Clearly not specific enough.