At this point, it's anyones guess, since you omitted the details of how you
backed up the old laptop.
In any event, backup the new one before you attempt to recover files from
the old laptops backup.
And make sure that you have a bootable restore CD, to provide a recovery
method if things go completely awry.
I'd guess that you have some sort of drive image, and "restoring it" to a
drive may wipe out whatever was previously on the drive.
Many years ago, I worked with HP minicomputer based "technical systems".
These systems used removable disk hard drives. We maintained three levels
of backup. There was a remote possibility that a backup might be destroyed
during a restore proceedure. Sure enough, I got a call one evening from the
night shift. They had had a problem, tried the first level of backup, which
failed. They then also consumed the second level of backup as well, and
wanted to try the third level. My answer was unprintable. It was obvious
that the cause of the problem had not been found, and the last level of
backup should not be used until whatever was wrong corrected. We had
diagnostics that were loaded from milar tape that were intended to be used
in such a situation before the second backup level was used, and they had
not been run.