Having them doesn’t provide any insight that the user needs. Your users couldn’t care less about what software you’re running behind the scenes.
Having them doesn’t allow “hackability” of the URLs. For example, look at the URL shanghaitang.com/shop/women/dresses.html. If a visitor just wanted to view the women’s page, hacking off the dress.html and going to just shanghaitang.com/shop/women doesn’t work. The correct URL is shanghaitang.com/shop/women.html. When a URL is several levels deep, users should be able to chop off parts at the end (“hack the URL”) and still be able to get to a usable page.
There’s a 35 character limit on the Display URL in AdWords, so the shorter the URL the better. URLs with extensions are simply longer URL. If you have long URLs, in AdWords you have to make up shorter Display URLs and when people click it they get taken to the Actual URL, but having the Display URL and the Actual URL the same would be great.
Easier to share and use in ad campaigns in general. “Toyota dot com slash camry” is much easier to say than “toyota dot com slash camry dot h-t-m-l“
By not having extensions, you’re future-proofing your site. If you have .html extensions today and convert your site to PHP tomorrow and all the pages get changed to end in .php, you’ve just broken all your links pointing to the URLs ending with .html, so it’s better to not have extensions at all.