I like the carving axe. I find the semi-bearded head allows you to get your hand close behind it, and the curved profile alows you to take out shaving instead of digging in.
Here is a picture of the granfors carving axe and the carpenters axe side by side.
I find the curved edge much easier personally than the straight edge. Maybe the straight edge of the carpenters axe would be better for straight lines in straight wood, but for carving through grain and not lifting splinters I think a thin finely bevelled curved edge gets my vote.
Most cheaper axes would need a fair amount of file work to get the bevel of the blade finer. The steel will be quite good, just the edge shape would need alteration.
Here is a bow I made with my carving axe. using it I could understand some of the old bows types found. When chopping out a bow and tillering it with a tool like an axe some limb shapes and profiles just come naturally.
Mark in England