As I understand it, round dancing is choreographed to a particular tune, with a main repeating group plus intro and breaks etc. A caller gives the cues for each figure as it appears.
Sequence is similar; no caller, and a fixed 16 bar repeated pattern which can be danced to any suitable tune. Around 30 to 40 new dances are published each year, and have been for about 50 years! They use standard ballroom and latin figures and variations. Classical (aka Old Time) is another style similar to what I think is called vintage in the US. This includes saunters, 2-steps (not cowboy), gavottes and medium speed waltzes with lots of rotaries and 3rds and 5ths.
Apart from the UK, sequence is popular in Oz and NZ.
For more info and a forum, see
www.sequencedancing.co.uk