The most obvious benefit I see, to convexing an otherwise 'flat' grind blade, is that the absence of hard, crisp shoulders behind the apex makes for much slicker cutting in thicker/tougher materials, like cardboard or wood. More than once, I've noticed how the crisp shoulders of a V-grind edge end can sometimes dig in and bind, in cardboard especially. You can actually see where the cardboard hangs up on the shoulders and deforms to the exact profile of the shoulders. A polished convex behind the apex, on the other hand, makes a world of difference in such materials, as it eliminates anything that can catch and bind, behind the edge.
I don't put a lot of stock into claims of a convex being more durable than a V-ground edge. To me, it's a moot point. The overall durability of the edge is basically all about the geometry at the apex, and the thinness of the steel behind it. Doesn't really matter what the shape of the transition between the edge and shoulders is, in that regard, other than the as-mentioned point earlier that the thinness of the grind will be changed one way or the other, in changing the shape of it from one to another. Making a V-grind convex, by knocking down the shoulders, inherently removes metal behind the edge, which makes it thinner behind the edge. By itself, there's no reason to believe it'll magically make the edge more durable, by thinning the steel behind it.
The apex should ALWAYS be essentially a V-grind anyway, at least to some microscopic degree, else it'll be thick-angled and dull if it's rounded ('convexed') all the way out to the apex itself.