Both corporeal and corporal have similar etymologies, and the difference seems to be very nuanced (see the fact that EO points us to corporality, meaning of corporeal existence).
From Etymology Online:
Corporal (adj.) - "of or belonging to the body," late 14c., from Old French corporal (12c., Modern French corporel) "of the body, physical, strong," from Latin corporalis "pertaining to the body," from corpus (genitive corporis) "body" (from PIE root *kwrep- "body, form, appearance"). Corporal punishment "punishment of the body" (as opposed to fine or loss of rank or privilege) is from 1580s. Related: Corporality.
Corporeal (adj.) - early 15c., with adjectival suffix -al (1) + Latin corporeus "of the nature of a body," from corpus "body" (living or dead), from PIE *kwrpes, from root *kwrep- "body, form, appearance."
If something was corporeal punishment, what would it be then? Or, would saying corporeal punishment be technically correct but only hit our ears wrong?
If they are different enough, what about the e makes this so?