Which is more gramatical and why?

Than me.

I is a subject pronoun. In this sentence i/me is the object

Both could be considered correct. The first part could be considered an elliptical sentence. In this type of sentence a word is omitted, but it’s easy to infer what the missing word would be:

Mary runs faster than I [do], but John is the fastest. The “do” is understood. It’s been omitted, but understood.

If the sentence goes on as in the example of above, you could leave it. However, if it’s a “stand alone”, you could also go with the object pronoun “me” because the word “than” is a preposition your noun must be in objective case (for it to be the object of the preposition).

You’re more likely to hear that the “me” version is “more correct” simply because most people aren’t aware of elliptical sentence construction (however, most English teachers know about elliptical sentences).

One caveat (and a personal pet peeve) this does not work with “Between you and I”—that is NEVER right even though newscasters frequently butcher this. It is ALWAYS “Between you and me”.

“Mary runs faster than I”, because if you said, “Me runs faster than Mary” it would make less sense than “I run faster than Mary”

TRUST ME!! its I I is a subject