A twitch is used to control the animal on the spot usually to administer vet care. A ring gives extra control but the animal can still keep moving. A twitch is used because an animal isn't cooperating. A ring comes into play when needed.
That is what I was asking in a nutshell, why does a twitch used on a horse work and not a ring in the nose as in cattle?
Sorry, I asked the first question at 2 am, last night was not a good sleep night.

Different animals are controlled in different ways and it seems really random and more culturally based than anything else.