UNIT is an international military organisation. As Maria Jackson told her father in "The Curse of the Berserker", "They're like the army, except they deal with aliens and things."
The Torchwood Institute was set up by Queen Victoria to defend Britain (and, originally, the British Empire) from alien threats -- one of which, in her view, was the Doctor. One of the ways the Torchwood Institute tried to do this was by scavenging technology from aliens who (either deliberately or accidentally) came to Earth.
So, besides the national vs. International character, UNIT is a branch of the army like the SAS is, Torchwood is more of a secret organisation like MI5.