Besides 따뜻하다 having an extra usage to describe personality, 따뜻하다 and 따스하다 both mean warm, at least that's what I found in my dictionary. Are these words interchangeable, or do they have nuances/different degrees of warmth, etc?
There are differences.
The difference is that 따뜻하다 is simply descriptive or tangible without emotions involved while 따스하다 is sentimental or intangible and is usually more emotional.
Example: You can say 마음이 따스하다, but seldom 마음이 따뜻하다 since 마음 is intangible.
Ex: 홍차가 따뜻하다 instead of 따스하다 ! since 홍차(red tea) is tangible.