This topic was brought up in one of the lesson commentaries, whereupon this link was presented:
http://home.alc.co.jp/db/owa/jpn_npa?stage=2&sn=68
It isn't of much good to those who can't tackle a somewhat abstract grammatical explanation in Japanese, so I'll try to boil down to the meat of it:
While both 気持ち and 気分 can both express one's physical state, on the mental side of things, 気持ち expresses a feeling based on an object or thing which is clearly represented, sort of stating a feeling based on things observed outside of the mind. 気分, on the other hand, refers to the psychological side of things, how
you, yourself feel in a situation instead of how something else makes you feel. The sentences used on the ALC page are these:
(round one, where either is fine)
A:具合でも悪いんですか。
B:ええ、ちょっと気持ちが悪くて/気分が悪くて。
guai demo waruin desu ka? (something got you down?)
ee, chotto kimochi ga warukute/kibun ga warukute (yeah, I'm a little out of it...)
(round two, featuring 気持ちが悪い, "feeling about something else")
殿様ガエルは鳴き声もグロテスクで、気持ちが悪い。(NOT: ×気分が悪い)
tonosama-gaeru wa nakigoe mo gurotesuku de, kimochi ga warui (NOT: kibun ga warui)
"The call of the bullfrogs is so grotesque, gives me the chills (the thing makes me feel bad)"
(round 3, talking on the inside, the expression refers to "me," my situation, and not specifically what it is that makes me feel bad)
人前でからかわれて気分が悪かった。
hitomae de karakawarete kibun ga warukatta
Being ridiculed in public is humiliating (it is psychologically hard)
However, the article ends with a modification of the bullfrog sentence to allow for 気分が悪い:
殿様ガエルは鳴き声もグロテスクで、聞いていて気分が悪い。
tonosama-gaeru wa nakigoe mo gurotesuku de, kiite ite kibun ga warui
The call of the bullfrogs is so grotesque,
hearing it (psychological) is nasty...
Whenever there's overlap things get confusing. I'd like to say I understand it perfectly, but I know I don't. This is the kind of thing that just develops over constant exposure and analysis.