Student or Learner
I am taller than him.
I am taller than he is.
Are these both correct?
I would say that both are equally correct in current use of English.
If you leave the verb in the sentence than you must use the subject pronoun, whereas if you leave the verb as understood then you can either use the subject pronoun (formal) or the object pronoun (informal).
Main verbs can be substitued with auxiliaries. The use of the object pronoun is much newer and probably was originally only a colloquial form but it is now deeply entrenched in everyday use and I don't think it can really be considered grammatically incorrect any more.
A further example to illustrate my point:
She works harder than I work. (repetition of work sounds unnatural)
She works harder than I do. (used a lot and sounds very natural)
She works harder than I. (very formal, sounds a little unnatural)
She works harder than me. (probably the most commonly used form)