It does have a few minor advantages: multiple attachment support, more "branding" images support (author avatars, post "banners"), easier to build JSON than XML in most API backends these days. The one major advantage is that JSONFeed is easier to work with it in tandem with WebSub and other pubsub systems that assume messages are natively JSON, you can use the same JSONFeed item generating code for both pull (HTTP GET) and push (WebSub) scenarios.

Certainly even the main advantage isn't a huge reason to switch to JSONFeed if you've already got RSS/Atom feeds, but I believe the hope for JSONFeed was always that it would spark some sites/backends that don't want to support XML, don't have good XML libraries, or don't want to use their template languages to drive RSS/Atom feeds to be able to build something simpler instead resulting in more feeds overall than if things were just left to the XML-ish status quo. I don't know how successful it has been on that front, though, but I appreciate it exists for trying.