Nutrition should be the same.
The home raised Cornish Cross are nothing like store bought birds. So you are way ahead on taste and texture, right off the bat. (if they are well raised and not raised like commercial birds)
The dual purpose birds will have stronger flavor and stronger texture because they are butchered much older. If you want that firmer texture and stronger flavor, you can raise heritage breeds or you can just run your Cornish Cross on to an older age.
A lot depends on which piece of chicken you like. My family likes breast meat and won't eat the legs, so the dual purpose or heritage birds would be a total waste for me. They also cost a lot more per pound to get to the table, and while I am willing to pay more than store bought, cost is a consideration.
I suggest that you buy a couple of Cornish Cross and a few different heritage breeds and raise them up and see which one your family likes to eat. Nobody on the internet can tell you which you will think tastes better.
