Generally Leafletjs and Mapbox.js are the same, but with Mapbox.js having plugins and extensions which wrap Leaflet and tie into Mapbox's services (eg directions). Similar plugins and features exist from other companies or products and Leaflet can use them in addition to, or instead of Mapbox. The Leaflet-based libraries typically have better legacy browser support, use raster tiles, etc. That said features are being added that incorporate modern features like vector tiles (pbf, mvt, etc) and various renderers (including webgl).
Mapbox-gl-js and the native variant mapbox-gl are open-source libraries that are high-performance, highly optimized around vector tiles (pbf, mvt) and webgl for rendering into a canvas element (for the -js variant). It is relatively new so some things that are easy with Leaflet might be different or challenging (as of April 2016), that said they're very similar and work quite well, including on mobile devices (from the past few years, eg iphone 5s). An example of a random quirk is that Hebrew labels in Israel, which read right-to-left, are backward and look like nonsense (it's an open issue being addressed).
If dropping older browser support is okay going the Mapbox-gl(-js) route can be a fine choice. In my limited experience (working with it a few months) it has the best user+developer experience overall and Mapbox has been consistent in their engineering/output. I have less experience with their paid services and it's unclear how tightly coupled their libraries will be to these services. For a mobile project I moved to mapbox-gl-js after looking at Google Maps, Leaflet v0.7 and v1 and it seems to have been a good decision.
I started using Mapbox-gl-js with prior Leaflet experience and proficiency in HTML/CSS/JS and found the primer and examples both helpful in understanding technical details. Including how the map is styled with JSON (not CSS). Also take a close look at the terms of service, this was an important positive differentiator especially compared with Google. Mapbox's services don't have the most robust coverage outside the USA so be sure to give this some review as well (in my experience another provider is typically available, so this doesn't necessarily affect the decision to adopt the libraries, they're just very tightly coupled to Mapbox features or standards).