Keep "internet" and "web" lowercase.
Every year at the American Copy Editors Society annual meeting, the Associated Press announces major changes to the AP Stylebook. Past bombshells include allowing email without a hyphen, allowing over to mean "more than," and allowing writers to begin a sentence with hopefully in the sense of "it is hoped that." These announcements can induce mild rumbling among the attendees and predictable freak-outs on Twitter.
AP Style Is Changing: Starting June 1, writers should no longer capitalize internet or web.
This year was no different.
Starting June 1, 2016, Associated Press writers will be instructed to lowercase both internet and web.
Although both the AP Stylebook and Chicago Manual of Style have long insisted on captializing internet (and had logical reasons for doing so), people on that loud and rowdy internet have long insisted on lowercasing internet. I have had many an exchange with writers who didn't care what the stylebooks recommended—or why—and felt deep, deep in their hearts that internet should be lowercase. I once had a writer email Tim Berners-Lee, the inventor of the web, to get a ruling on whether web should be capitalized. (Berners-Lee said lowercase. I stuck with "AP says uppercase.")
Given the emotion and insistence I've encountered about lowercasing internet and web, I'm not surprised that the AP editors changed their style. It will be a relief not to have to fight about it anymore.
Now the only problem is that AP and Chicago disagree, so you'll see it both ways for a while in edited text, and freelance copy editors will need to remember which style guide their clients prefer.
UPDATE March 23, 2017: Chicago just announced that its forthcoming 17th edition will lowercase internet.