There is actually a distinction.

I raise hair sheep (Katahdin/Barbados crosses) , which produce no useful (that I've seen) wool. But are fantastic in terms of meat.

I'm starting to see a market for them in the specialty realm (beyond halal/kosher/immigrant markets)

I'm thinking I'll need to eventually set up something like this: https://www.deepsouthventures.com/i-sell-onions-on-the-inter...

In Greece where we regularly consume sheep it's mostly both, mainly because we don't have specialized breeds for wool. They do exist, but they're rare.

Just to add to the mutton pool, you also have old female sheeps that are no longer able to give birth. There's a chap in Cornwall adding value to these old ladies by giving them a bit of a retirement, getting them fit and healthy before slaughtering and dry aging them for some top UK restaurants. Check out The Cornwall Project - Matt Chatfield. Bit more about it here - https://www.scottgrummett.com/blog/the-cornwall-project. I am not affiliated, but would like to try it someday.

I can't speak to sheep specifically, but we use different species of goats and cattle and chickens when raised for meat vs for dairy or eggs.

Still, dairy cows do eventually become beef. If you ever see a cut of beef that's normally bone-in but the butcher chose to debone this one, then there's a good chance it was from a dairy cow whose bones were too calcified to saw through.

Those who raise sheep for the wool usually raise expensive ones with pedigree like the merino breed in Spain.

I was under the impression that lamb was from young sheep raised explicitly for meat, while mutton was from old sheep after their wool career was over, but I can't say I've investigated it deeply.