Moving Drupal to a new host but same domain
Few things to check.

Which version of PHP? I've seen Drupal sites break going from 5.3 to 5.4

You might want to check what the TTL of your domain is. If it's long you'll have people visiting the new site and the old one at the same time. You can set it to 5 minutes (300) which should make sure everbody moves across to the new site at roughly the same time.

Stress testing. Can the new hosting actually handle the expected load? Things like uploading images will take a lot of memory.