On the one that isn’t working the page has no sub-title (nothing before the “» Southampton Climbing Wall”), and that the code is somehow inserting and tags in random spots.
That’s right. The text in p.page-banner-heading (where it reads Intro to Climbing Course – 1st July, 6pm in the working example) is also absent and that’s likely to be causing a knock on effect, with everything else displaying higher up than it should.
inserting p and break tags in random spots
This can be caused by other plugins but usually is because the theme is applying wpautop or other formatting measures later on than usual – and, unfortunately, that can have this peculiar side effect where content from plugins like The Events Calendar becomes peppered with paragraph and break tags.
Tim: have you tried switching the template in use via Events > Settings > Template?