I know everyone's experiences and preferences are different. I have both Jerseys and Nubians, and I prefer the Jersey. My Jerseys produce a lot more milk, the cream can be skimmed off the top to make butter/whipping cream, etc. And I feed the cow as much as the goat, who produces significantly less. That said, the cow is much more economical for me.
The goat milk is just as good, but you need a cream separator because goat milk is naturally homogenized. So it doesn't separate well. You can still make great cheese with either.
I find it's much easier to keep the cow in condition, but have to work a bit harder to keep the goats in condition.
My cows are much easier to keep contained in their pasture (two strands of electric), and are less prone to predators (where I live!) than the goats. Some goats are difficult to contain, they need to be wormed much more often than a cow.
As for handling, I think both are just as easy to handle. All of my girls are well behaved on the stanchion.
Pecorino Romano cheese is made from sheep milk
That's the good stuff!
And just to give you more ideas, traditional mozarella is made from buffalo milk and I hear they're wonderful on the stanchion as well
Though you can make it with goat milk, a combo of cow and goat milk, or just cow milk.......or buffalo milk!