When you use a pronoun in a comparison using the words than or as, use the proper pronouns as if all the words were being said.
Most of the time when we use a comparison using than or as, we leave words out. This is technically called an elliptical clause--a clause with an ellipsis. An ellipsis is words left out.
Look at it this way. There is a difference between the two following sentences. Both are grammatically correct; they just mean two different things.
He likes you more than me.
He likes you more than I.
Think of what words are left out:
He likes you more than I do.
(I is the subject)
He likes you more than he likes me.
(Me is the direct object)
When a pronoun follows than or as in a comparison, make sure you understand what words are missing and then use the correct pronoun.
Incorrect: He is taller than her.
(i.e., than her is?)
Correct: He is taller than she.
(i.e., than she is. Much better!)
Incorrect: He is as happy as them.
(i.e., as happy as them are?)
Correct: He is as happy as they.
(i.e., as happy as they are.)
Correct with one meaning:
He sees you more often than I. (i.e., than I see you.)
Correct with another meaning:
He sees you more often than me. (i.e., than he sees me.)
The case of the pronoun makes the difference!
For more, see Pronoun Case.