Both types force food through a perforated screen and what makes a difference (for the food) is the size of the holes.
The conical version (bottom picture) can process more food before the screen is fully clogged (more screen surface), and usually you can just pop out the cone and empty the skins and seed. Whereas with the traditional food mill you have to disassemble. With some foods (whole tomatoes) you may have to pass them a couple of times through the mill to get everything.
Why you might choose one over the other:
If you are processing bushels of stuff, I wouldn't even think of the traditional food mill. We use a 1/2 horse power belt driven spiral sauce maker and it already takes us hours to make ~200 jars. I'd become a zenist if someone handed me a traditional mill with bushels of tomato.
For home use and cooking for under 20 people the traditional mill is great since you can just place it over the pot, go, and throw in the DW. No mounting fuss, and fits into a normal cupboard. You CAN find decent Italian made ones for $20 in North America.