"I do not feel well today."
Translation:Mi-e rău astăzi.
If I understand your question correctly (E3L speaker here, don't know what to make of is+are), I don't think they mean the same.
In my opinion, “mi-e rău” is stronger than “nu mi-e bine” or “nu mă simt bine.” But obviously the author of the default solution does not agree with me. (Lurch_M, however, seems to agree ☺.)
Edit: Thanks to Lurch_M for pointing out the emotional meaning of “to feel bad (about something),” translated by “prost.” I am no longer sure whether Lurch_M agrees to my idea of a “neutral zone” between bine and rău. In my language I might feel not really good without feeling really sick. Maybe in Romanian, or in English, one feels sick (rău) as soon as one is no longer feeling perfectly well (bine).