The cgi should be though you may have to change the ownership of them if you end up with a different username. webmail may get added to the backup. Not entirely clear on whether the cpanel backup tar's /home or just /public_html. Generally when people e-mail us in regard to backup issues they've hosed their website and we pull it from our backup server rather than anything cpanel does.
Personally, I'd just pack the website rather than mail as well. E-mail's such a touchy system it would be best to just download the important stuff, then set up mail again on the new account fresh.
If you have shell access you can do a backup and transfer easily enough.
tar cf *filename*.tar public_html
and download it (or wget it from shell on your new host)
Once it's at the new host:
tar xf *filename*.tar
ls -la <-- check permissions.
That'll unpack and overwrite the public_html folder. Everything should remain as it was, including file permissions (755, 777 etc)
As far as I can tell that's pretty much what the cpanel backup system uses anyway - it's just a manual version of it.
Former Webhost... now, just a guy.