A convex grind is going to have more steel and thus more strength behind it. The down side is that it will have a steadily increasing angle as you get back from the edge which will make it harder to slip into the cut. It will have to push the two layers apart more. With a flat primary grind and a convex secondary grind you limit how far back on the edge you have a problem with that at the expense of more strength in the blade with a full convex grind. The full convex grind will make a less handy slicer than a flat grind but it will be a stronger chopper. A flat primary grind with a convex secondary grind or a convex primary grind with a flat secondary grind will be a compromise between the two.
To illustrate another issue, I made two knives from 9260 and hardened then tempered them together. Both had flat primary grinds. One had a convex secondary grind and the other a flat secondary grind. The one that had the convex secondary grind passed the wire cutting test with no more than a slight dulling of the edge. The one with the flat secondary grind indented with rolled over edges indicating too soft a temper. I at first thought that I hadn't properly hardened the blade before tempering and I went back and repeated everything but I got the same result when I tested it the second time so I went back, again, and rehardened the blade and then tempered at 25 degrees lower. It then passed the wire test. It turned out for that steel with that grind I needed a higher hardness or I needed more steel behind the edge with a slightly softer temper to prevent failure of the edge.
I could have also found that the convex edge might have resisted chipping out better than a edge with a different knife due to the added strength behind the edge. But that would have been a some expense of it's slicing ability. When it comes to knife design, everything is trade off and compromise.
Doug