A big difference indeed!
Is a cloud hosted email system independent of your hosting account, CloudMail accounts allow for a lot (5GB -15GB) more storage and have many more features than Local mail accounts.
Local mail is handled and stored on your local web server, all storage for the email users counts as part of your disk space usage. If you have a small website but a lot of mail users CloudMail will be a better option for you.
How do I tell the difference?
Log into your account center and look at your service, if it says "with xx cloudmail accounts" then you are using CloudMail. If it does not say that you will be using local mail.
Just ask, call or open a support ticket.