What is the difference between exhausted and enervated when used to refer to human beings?
There is a biochemical and anatomical difference. Exhausted would implies that there is no available energy for the task at hand and enervated would suggest that the intended task lack the nervous potential to be actuated. That said, in popular discourse, there is not much difference between the two.
The 'exhaust' meaning of the word enervate is a figurative sense that came in later.
The original, now archaic/DSL meaning of enervation is to 'cut the nerve (of)', early 15C. The figurative sense is from the 1550s.
enervate: (medicine) To partially or completely remove a nerve.
Today, enervate is used in formal or literary writing and in medical literature. Exhaust and its many synonyms are preferred in general English writing and speech. Note that in medical writing, the word may be used in either sense, so care needs to be exercised in drawing the implication in a context.
Use enervate, not exhaust when you mean a figurative: To deprive of nerve, force, strength, or courage; to render feeble or impotent; to make effeminate; to impair the moral powers of. (various dictionaries)
Alec Baldwin: Now that this enervating election is over, let's help rebuild New Jersey, Staten Island, the Rockaways, et al.
You will also find enervate used as a synonym of exhaust in the writings of respectable authors, though.
Use exhaust when you mean literal and in informal writing.