The two verbs réveiller and éveiller. Both are commonly defined to mean "to wake up" or to "awaken", and likewise the adjectival forms (i.e. participles) réveillé(e) and éveillé(e) both often are defined to mean "awake". Is there any difference between how I'm supposed to use these two words, or are they truly identical?
They can be interchangeable at times, not always. Here's a good attempt at defining the difference :
(Very approximative translation: Éveiller is to draw someone out of their sleep, réveiller is also to draw someone out of their sleep, but with uncommon effort to break sleepiness.)
I think in all literal senses réveiller could always be used, but not éveiller. I would say indifferently: « je dors mal, je me réveille souvent la nuit / Je m'éveille souvent la nuit ».
But I would say un bruit à réveiller les morts ("a noise that would awaken the dead"), and I would not say éveiller here.
Éveiller will be more easily used in the figurative sense. See the song Il est cinq heures, Paris s'éveille. And in that example, it gives a poetical touch that réveiller wouldn't have.
We'd say of a child « il est bien éveillé pour son âge » ("He's bright for his age") and we would not say that he's « bien réveillé » in that case.
(Réponse qui est loin d'être complète, la question a éveillé en moi des tas de réflexions que je vais ruminer. This answer is far from complete, the question engendered lots of thoughts that I have to ponder about.)