restore app from a different laptop but the same backup

I recently bought a new Macbook and new iphone. Before I abandoned my old thinkpad, I backuped my iphone and copied my iphone backup data from "C:\Documents and Settings\USERNAME\Application Data\Apple Computer\MobileSync\Backup" to the right location on my new laptop. So when my new iphone connected to my new laptop, I can restore it from backup. The problem is I only got my contact, sms, photos, notes and email back but not applications.

Because I used an iphone app that stored passwords for me, I really want to get the app data back. If I download the app again from app store, will it have my old data on it? Or are there any software that can restore the app data from my backup file?