Google Inbox is now generally available without any invites. However, the Chrome Inbox App is still alive and maintained (last version was February 9th).
Does the Chrome extension provide any extra features that do not exist in the inbox web site?
There is no difference at all.
The screenshot doesn't correspond to an extension. Please note that the right up corner only display the "Visit website" button but not the "Add to Chrome" button. The screenshot it's about the Inbox by Gmail directory/catalog entry about in the Chrome Web Store.
At this time the Chrome Web Store include four categories apps, games, extensions and themes and two types Chrome apps and websites.