Can I copy and paste the old database sql script on the new one ?
Actually, there's no need to delete the old website at all, as of v2.2.1-3, there's an additional file named manual_upgrade.php in the /install folder. Open that file and there's an instruction (commented out). Even if you copy and paste the old database SQL script, you will still need to run manual_upgrade.php (or fixupgrade.php in that documentation)
If you delete the old website, any link in your theme with old references to /js/bootstrap3 must be replaced to /skins/bootstrap3 if you use the built-in boostrap3 files.
1) Copy both manual_upgrade.php and upgrade.php from /install directory (in the schlix-cms-v2.2.x-y.zip ZIP file) to the root directory of your website where the file runtime.inc.php and index.php resides
2) Manually run http://your-website-address/manual_upgrade.php