"I am tired and ill."
Translation:Fáradt és beteg vagyok.
There is a grammatical difference. Rosszul is an adverb ("badly" or "poorly"), and beteg is an adjective ("sick" or "ill"). This has some consequences for how you use them. For example, rosszul must occur with a verb. And beteg like any other adjective can take plural, accusative, or other endings as if it were a noun.
And the meaning is different:
"Rosszul vagyok" - I feel sick, I am sick, I am not feeling well
"Beteg vagyok" - I am actually ill, lying in bed at home, caught a disease, etc.