Bristol University is like the Durham University of the south is my impression.
Exeter University's roots actually go back to the 1800s and it became a university in the 1950s so it doesn't fit in a particular box. Similar kind of story for Leicester University.
To the layperson, Bristol would be more prestigious than Exeter, perhaps largely because Bristol has more dramatic architecture, higher entry grades, and is regarded as a more studenty city than Exeter, but it also happens to have a slightly better research rating than Exeter.
Exeter has some old buildings amongst its modern buildings but it seems to generally model itself around a modern campus university feel.
Personally, I'd consider Kings College London or Nottingham too, for similar grades, because those universities look beautiful too.