Let me rephrase a few things I said. First I think clean code is very important, though the importance is more to do with development and maintenance and usability than it is to do with SEO. All of the above overlap though so clean code can improve SEO to a degree. Usually with SEO it's more that poor code can hurt you than clean code can help you.
As an example say your home page navigation is created with Flash and you have no other links on the page. Search engines don't really read Flash well so odds are they may never index any page beyond your home page. Fixing that by either changing the navigation or creating an alternate html navigation for spiders would seem like it improved your optimization, but again it's more that the Flash navigation was hurting you than the html navigation is helping you. There is a difference.
Steve with the site you gave as an example I think it's hard to determine what changes had the effect of improved ranking. You changed a lot of things and any or all could have had an effect. Most likely the increase in ranking was due to a combination of things, but it's not proof that any one change you made had an effect. You could have done everything with the exception of converting the layout from a table to css and might still have noticed the exact same results.
Bill the older sites ranking well is most likely that they had an advantage in quantity and quality of links due to being online as long as they have. It's possible that the age of the site played a role, but it's more likely the age of the links is why they rank so well.
When it comes to css vs table layout I don't see any evidence that one or the other is better for SEO. I wish the evidence pointed to css, but it doesn't. Search engines want to rank content. When comparing pages for ranking they looked at the content stripped of all the code anyway. I can offer you lots of reasons why you should use css over a table based layout, but SEO isn't one of them.