Excuse me for posting an answer without referring to any sources...
We usually use 理科 to refer to a school subject (which usually includes these 4 fields: physics, biology, earth science and chemistry). We usually use the word 理科 in upper grades at elementary school, and in senior high we have [理科第1分野]{りかだいいちぶんや} (=chemistry and physics) and [理科第2分野]{りかだいにぶんや} (=earth science and biology). In senior high we usually choose 2 to 4 from these 4 fields and don't use the word 理科 anymore. I think we sometimes call these 4 subjects '[理科系科目]{りかけいかもく}...maybe it's like 'the group of scientific subjects'/'one of the scientific subjects'.
And as Paul-san said, we often say 'She is 理系/あの子は理系だよ。', 'I am 文系[ぶんけい]/私、文系なの。' when we talk about which field of 理系 and 文系 someone majors/majored in at college, and also 'あそこの大学は文系だ/That university is 文系', 'ここの[学部]{がくぶ}・[学科]{かっか}は理系です/This faculty/department (at college) is 理系' etc. to talk about what kind of courses a college/faculty/department has.
As for 科学… sorry Dave-san, I still don't know what to write about it here...