HTML code errors & SEO

HTML Code Errors & SEO - Does Spelling Count?

I have received this advice:

To improve your search engine ranking, I would start by cleaning up your code. Your HTML is a mess. Take a look:
http://validator.w3.org/check?uri=http%3A%2F%2Fwww.Time4Learning.com%2F&charset=%28detect+automatically%29&doctype=Inline
It's important to have code that validates!

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I've checked my site (using DreamWeaver's tool and a few free website checks including the one above) and most pages have dozens and dozens of errors. I know that google says that they care about errors, what do you think. Does it make a tangible difference?

And if so, does the Dreamweaver function for automatically fixing errors work or do I need to do it by hand, page-by-page. Does anybody know someone who specializes in doing this?
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More data:

I have started with google to register for a site map but only got as far as validating my site which has earned me access to some new data.

They have some info on site errors including HTTP errors. The indicator shows only a tiny shading and the only listed error is a 404 with the URL (which doesn't exist: www.time4learning.com/om/alpha/Signup_Parent_Information.aspx .

Oddly enough, I am not doing well on the terms that I most care about such as homeschool curriculum . Stay tuned.....

I've put this out for discussion at the Dev Shack. http://forums.devshed.com/search-engine-optimization-108/html-code-errors-seo-361845.html#post1529955

Well, nobody at DevShack seems to have responded but a newsletter from EntireWeb written by the very organized Adam McFarland arrived on 7/4 answering my exact question.

ps - i updated my thoughts on html correctness on feb 07.