"You would not believe the sort of weird, random, ill-formed stuff that some people put up on the web: everything from tables nested to infinity and beyond, to web documents with a filetype of exe, to executables returned as text documents. In a 1996 paper titled "An Investigation of Documents from the World Wide Web," Inktomi Eric Brewer and colleagues discovered that over 40% of web pages had at least one syntax error".
We can often figure out the intent of the site owner, but mistakes do happen.
The number of webpages with HTML that's just plain wrong (and renders fine!) is staggering. I often wonder what the web would be like if web browsers threw an error upon encountering a syntax error rather than making a best effort to render.
The downside is maintainability. If your website follows the rules, you can be pretty confident that any weird behaviour you see is a problem with the browser (which is additional context you can use when googling for a solution). If your website requires browsers to quietly patch it into a working state, you have no guarantees that they'll all do it the same way and you'll probably spend a bunch of time working around the differing behaviour.
Obviously, that's not a problem if you already know exactly how different browsers will treat your code, or you're using parsing errors so elemental that they must be patched up identically for the page to work. For example, on the Google homepage, they don't escape ampersands that appear in URLs (like href="http://example.com/?foo=bar&baz=qux — the & should be &). That's a syntax error, but one that maybe 80% of the web commits, so any browser that couldn't handle it wouldn't be very useful.
It's interesting that you apparently actually checked a validator to get the error count, and yet the two things you cite as errors are not errors, have never been errors, and are not listed in the errors returned by the validator. Both opening and closing tags for the html, body, and head elements are optional, in all versions of HTML that I am aware of (outside of XHTML, which has never been seriously used on the open web as it isn't supported by IE pre-9). There is a tag reported unclosed by the validator, but that's the center tag.
Anyhow, one downside to having syntax errors might be that parsers which aren't as clever as those in web browsers, and which haven't caught up with the HTML5 parser standard, might choke on your page. This means that crawlers and other software that might try to extract semantic information (like microformat/microdata parsers) might not be able to parse your page. Google probably doesn't need to worry about this too much; there's no real benefit they get from having anyone crawl or extract information from their home page, and there is significant benefit from reducing the number of bytes as much as possible while still remaining compatible with all common web browsers.
I really wish that HTML5 would stop calling many of these problems "errors." They are really more like warnings in any other compiler. There is well-defined, sensible behavior for them specified in the standard. There is no real guesswork being made on the part of the parser, in which the user's intentions are unclear and the parser just needs to make an arbitrary choice and keep going (except for the unclosed center tag, because unclosed tags for anything but the few valid ones can indicate that someone made a mistake in authoring). Many of the "errors" are stylistic warnings, saying that you should use CSS instead of the older presentational attributes, but all of the presentational attributes are still defined and still will be indefinitely, as no one can remove support for them without breaking the web.