Before I start blabbin', let me just say I'm a student of this stuff, not an expert. My opinions are based on my own experience and I could be totally wrong.
Convex grinds... it depends on how you how you define one, and how it's applied. To many folks, simply feathering back the original edge a bit makes it "convex." This is true, sort of, since you no longer have a V at the edge, more like the thin end of an appleseed. On the other hand, a full convex grind makes a full, smooth sweep from edge to spine. If you really wanted to, you could say a piece of roundstock has a convex grind, but that would be silly.
The first example may hold its edge a little longer, it may not. I have noticed this in my knives, but I can't quantify it. They held an edge pretty good beforehand.
It may cut more smoothly, this seems reasonable to me and has been my experience. No hard, crisp angle to catch on. A hollow grind cuts even more smoothly, for the same reason, less drag. Until you hit the flat of the blade.
Sharpness won't increase unless you decrease the angle you sharpen at and/or increase the polish of the edge bevels. I don't care what style of grind it is.
With a full convex grind, the advantages are more pronounced. But only if the blade is wide and thick enough. I think drag/binding is the biggest factor at play here, as Magnum22 said. That's not much of an issue on a Mora. On a 1-1/2 wide, 5/16" thick Bowie, it definitely can be an issue. There is a maker here who also regrinds (usually large) production knives to a full convex, and his customers rave about the increase in performance.
also some find it relevant that as you sharpen a flat knife you gradually move back toward the spine and get a thicker edge. convex sharpening supposedly keeps the edge consistent. i don't quite buy any part of that one.
I disagree, sort of... of course you can keep the cutting performance more consistant with a convex grind, if you're really keeping that same sweep/arc as you get into the thicker part of the knife. If you sharpen normally, that edge (which is more obtuse than the blade grind) will get wider and wider. Draw a 40 degree angle (pretty standard sharpening angle) on a scrap of paper, and cross it where it's 1/8" apart. Doesn't look so sharp now, eh?
But I don't quite buy it either, because if you have to sharpen that far into the blade of your knife for it to make a difference, you either use it a HECK of a lot or the heat-treat is way, way off.
I like convex grinds, full and partial. I put them on the knives I like best and use most, and they perform better. Especially the big ones, they certainly chop and split wood better. But it's not a magic bullet by any means.
The cons are, you will scuff up the finish on your knife.