Also be careful about your terminology and what you mean - a convex edge, or a convex primary grind?
Or the full convex zero edge, which smoothly blends a convex edge into a convex primary grind.
There are lots of variations of all three - the NMSFNO CG came with a convex primary grind with a V ground edge, while the LE version of the same knife was a full convex zero edge. The Swamp Rat Vex is a flat primary grind, with a convex zero edge that's blended smoothly into that full flat primary grind (maybe the primary grind has a slight convex, but it's not much). Or you can reprofile the edge on a saber or full flat ground knife to have a convex edge, like I do on all my blades when I sharpen them.
Lots of variations, and it can be confusing unless you are very specific about your meaning when you talk about them.
As far as benefits - convex edges are very robust, and when blended well into the primary grind, they have no corner to bind up so they slice and chop very well. Convex primary grinds are awesome for chopping, since they don't get stuck in the wood. Their performance batoning isn't as noticeably different from a full flat grind, but they are WAY better than a saber grind, which can bind up like crazy due to the sudden transition where the saber begins.
Clear as mud?