CentOS is the Enterprise version of Red Hat Linux. The advantage to an Enterprise version is a longer life/upgrade cycle of five years (compared to other Linux distros that can be as short as 18 months). Why this matters is because though minor OS updates can be done in place, major version updates can not (or more accurately, it's strongly recommended against doing so).
The only noticeable change will be that because we'll need to migrate to a new machine, the server IPs will change. Therefore any IP-based usage you'll need to edit the IPs you're using. After all accounts are migrated over to the new server we'll update the nameserver IP, so the DNS transition should be smooth and uneventful.
All accounts (website files, email) will stay on the original server until they have been fully copied to the new server, tested for accuracy, and the DNS has finished propagating. Because of this we anticipate no downtime associated with this upgrade and account migration.