manifest.json vs manifest.webmanifest

There are no big difference rather than naming. Use whichever you prefer: "/manifest.json" or href="/manifest.webmanifest". Just don't forget that MIME type has to be: application/manifest+json in both cases.

<link rel="manifest" href="/manifest.json">
<link rel="manifest" href="/manifest.webmanifest">

Note that spec says it should be manifest.webmanifest. And later it may be important because it is under heavy development.

For more info check: W3C and MDN

November 2019 Update

The difference is mentioned in MDN (Mozilla Developer Network):

The .webmanifest extension is specified in the Media type registration section of the specification (the response of the manifest file should return Content-Type: application/manifest+json). Browsers generally support manifests with other appropriate extensions like .json (Content-Type: application/json).

You can read more details on: Media Type Registration

Like mentioned in the other answers, there's not much of a difference in both.
But, according to webhint.io, it is recommended to use manifest.webmanifest.

manifest-file-extension warns against using non-standard file extensions for the web app manifest file.

While the .webmanifest file extension is not enforced by the specification, nor is it required by browsers, using it makes it:

• easier to set custom server configurations for the web app manifest file
• possible to benefit from existing configurations

This is also displayed as a tip in Microsoft Edge Browser Dev-tools in the issue panel (screenshot below)