Q:

What does “not as [adjective] as him” imply?

If it is written that X is not as tall as Y, it indicates that X and Y have unequal heights. But does this necessarily imply that X is shorter than Y?

A:

In the absence of further elaboration, yes:  "I am not as X as you" implies that you have more X-quality than I do.  If X is "tall", then I am shorter than you.
It is possible to follow up for a humorous (or insulting) reversal of the normal implication:

I am not as good as he is... I'm actually way better.

But if you don't do this explicit countering, the default reading is that "not as X as" is always "less X than".

