I think there is a slight difference in meaning. "Jag mår illa." - I feel nauseated, like I might throw up. "Jag känner mig illa." - I don't feel well/weak/ill.
It possibly corresponds to the German expressions "Mir ist schlecht." (jag mår illa) and "Mir geht's schlecht./Ich fühle mich schlecht." (Jag känner mig illa)
Warning! In British English sick is not at all ambiguous when used with the verb feel. I feel sick/I'm feeling sick means I feel nauseous and I may be sick ie vomit at any moment... If you feel unwell in a different way, you might say I feel ill/I'm not feeling well/I'm feeling a bit rough.
As far as I know, "må illa" is used specifically for talking about feeling nausea. If I hear "I feel bad" that could mean that I am unhappy or unwell in some other way. For me "I feel sick" means that I am nauseous but "I am sick" just means I am unwell in some unspecified way.
Native speaker here. This is generally right on. (I upvoted you). Basically, 'feeling sick' is a little ambiguous. 'Feeling sick to your stomach' makes it clear that you're referring to nausea, although that's a little ambiguous too, because you could be meaning a stomachache. 'I am sick' is also ambiguous.