Well, it's true that "dubious" and "doubtful" can be synonymous. It's even true that they can be synonymous and hence interchangeable both when expressing the notion (when applied in relation to a person) that that person is to be doubted, and when expressing the notion that that person is the one doing the doubting.
But interchangeability breeds ambiguity, and best practice is to avoid it, and so it is good to favour one word for one context, and the other for the other. I believe common usage does prefer "doubtful" for the situation more accurately described as questioning (because it aligns well with "full of doubt"), and "dubious" for the other, more accurately described as questionable.
"Dubious" sounds more sexy than "doubtful", but I refer you to the entry "working and stylish words" in Fowler's, which, though not specifically contrasting "dubious" and "doubtful", does mention "dubiety" and "doubt".
Sexy is bad.
I shouldn't need to tell an a.u.e stalwart this. :-)