I don't really think his delivery was very awkward. If you watch, he was going for the more standard comic routine. Tell a joke, wait for laugh. Repeat. While I'm not saying that Louis CK is Johnny Carson, though I love them both, his cadence is quite similar to how Johnny would present his monologue. Many comics of the day had a similar style of comic presentation. (He probably also had a little bit of nervousness because he was on Letterman for the first time).

A lot has changed in the world of comedy in the past 14 years (Hell, whatever Comedy Central was called back then had virtually no content[it went through a few variations and I'm not quite sure when Comedy Central came into being]. Those were the days when all you could see on the channel was standup from comedy clubs). Louis CK not withstanding. He has changed the kind of comedy he tells, therefore his presentation has changed.

Today, Louis has become the story-teller comedian. During the early 90s, this kind of comic was not as common (Larry Miller is the only that comes to mind at the moment). Today, this kind of comic is quite common. Anyone telling a story is going to appear smoother and will captivate the audience more.

I think it was just a different time and the style of comedy has changed.