Éprouver sounds a bit more sophisticated and possibly less common. You can express a range of feeling with it: J'éprouve du chagrin (sadness). J'éprouve de la difficulté (I have some difficulties). J'éprouve de la douleur (pain). J'éprouve de la tendresse (tenderness). Sentir is both to feel and to smell. A bit more tricky, as once a student told me "Je sens très mauvais" (I smell really bad) when she meant to say "Je ME sens très mal" (I feel really bad). So for feeling, it needs a pronoun! "se sentir". I usually use "se sentir" only when giving a very vague answer: "je me sens bien (good). je me sens en forme (in shape). Je me sens mal (bad)." I would rather use real feelings directly otherwise, like "je suis triste" (sad) "je suis fatiguée" (tired) but there is no rule saying you can't say "je me sens fatigué" or "je me sens triste".

pour moi, feel = sentir, par exemple, I feel good = je me sens bien. je sais pas comment utiliser eprouver...