"My vocal cords hurt"
"My voice hurts"

these two sentences have the same meaning?
If not, could you let me know what each sentence means?

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"My voice hurts" would be more of a slang expression than a grammatically correct sentence because "voice" is not something physical that can hurt... It's a sound you make with your vocal cords.

"My vocal cords hurt" is grammatically correct, but not commonly heard because it's too technically detailed, that is, more of an anatomical phrase.

It would be more common to hear "My throat hurts" even though it may really be your vocal cords that hurt, rather than the throat itself, but this would be a common phrase to hear if someone was hurting from shouting or singing too loudly.