Though the validator is irrelevant to how a browser renders your code, it’s a great way to find problems.
How is it irrelevant? Although a lot of of the info on there may be useless to most, I would say pointing out invalid characters in your doctype, and missing closing tags are quite relevant to how a browser renders code.
Especially in the case of IE, the validator will tell you exactly what is wrong. 9 times out of 10 when looking at websites, issues with the display within IE have everything to do with the code, and very little to do with the css differences.
The other browsers all ignore that stuff…. but not good ole’ IE
It’s just important to understand that the validator is a troubleshooting tool. A page can work just fine with validation errors.
Run my semantic html5 pages through that validator, and you’ll get a ton of errors. The validator has no idea that I use Modernizr, and doesn’t recognize html5 elements. That doesn’t make my code wrong, or make IE render it differently.