There is no official interaction. UNIT stands for the United Nations Intelligence Taskforce. While they are dominated by Britain because, let’s be honest, Doctor Who is a BBC show, they’re not a British organization.

Torchwood IS a British organization, set up and paid by the British crown (not even the government). They’re also highly classified, so for the most part, UNIT doesn’t even know about them. The higher ups in UNIT have probably been informed of or discovered Torchwood’s existence, and might call them in when they have no other options; but in general, they are competing, not collaborating, agencies.

As far as differences in focus, UNIT is more of a police organization; their main mission is to serve as the contact point for responding to alien communications and to contain alien incursions.

Torchwood’s focus is more on obtaining alien technology, studying it, and using it for the betterment and defense of the British Empire.

In the second doctor episode that the Organisation was still called UNIT but the acronym meant United Nations Intelligence Taskforce now it means UNified Intelligence Taskforce. What is the reason for this. basically real life interacting with fiction. The real reason was that the real life (? – they still seem to live in a rarified bubble thinging they are more important than what they actually are!) UNited Nations didn’t want to be associated with an alien police force! Hmm, lot’s of lines that i will leave alone in case I join the queue that have offended António Manuel de Oliveira Guterres!