You're absolutely right to point out the distinction — and you're getting into the heart of prescriptive grammar vs colloquial usage.
Let’s break it down:

Grammatically Correct (Formal/Prescriptive):
"Nobody knows this better than I do."
This is a complete sentence, where "do" is the elliptical verb standing in for “know.”
“I” is the subject of the implied verb “do (know this).”
This version is grammatically impeccable and preferred in formal writing or academic contexts.

Technically Incorrect (but Common in Spoken English):
"Nobody knows this better than me."
Here, “me” is used as the object, even though it’s functioning as a subject of the implied verb.
While it's not formally correct, it's extremely common in spoken English, especially in informal settings.