In "not as intelligent as I" we're seeing ellipsis at work: "I" implies "I am."
In "not as intelligent as me" you've encountered something that gives some English "grammarians" conniptions: the oblique pronoun. I'll give you a couple of other examples with "me."
There's the old "it is I/it is me" argument. But "me" seems to have won this battle; when someone says "It is I" these days, it tends to sound rather melodramatic and affected.
Then there's "Jim and me went to the movies." This usage of "me" - even though millions of native speakers routinely resort to it (and have done so for centuries) - is still frowned upon. Like "ain't" it remains a bit of a Rodney Dangerfield in English.
'not as tall as me' and 'not as tall as I' are both used. It is proper to use 'not as tall as me' because the last 'as' is a preposition and should be followed by the object form of a noun or pronoun. 'I' is the subject form and when you use it it should be followed by a verb, for example 'He is not as tall as I am'.
There is debate about whether it's a preposition or a conjunction, and you don't need to follow 'I' with a verb- 'as tall as I' is perfectly acceptable. There are three ways of handling this:
as tall as I
as tall as me
as tall as I am