I don't think there are real differences between the terms. Usually people are using these terms to refer to the same thing.
In South Africa we use the term “women's budget” because we have 11 official languages, and only one of those languages, which is English, has the word “gender” in it. So to make things simpler and clearer to everybody, we called it a women's budget. In retrospect, it was probably a mistake in that it made people think it was a separate budget for women.
My own preference is “gender-responsive budget”, because that's what we're aiming at. But I don't think the different terms are all that important.