iDevices can only be synched with one library. However, you can sync successfully with more than one computer if each has a copy of the same logical library - the identifier that iTunes uses to determine which library is synced an iDevice is recorded within the iTunes database (the iTunes Library.itl file). The best approach is to replicate your complete library (database, media, artwork, associated files) from one PC to the other. When you reinstalled iTunes on your laptop it will, by default, have created a new library with a distinct id from that of your other machine. Even if you copied all your media from one to the other, iTunes will still see the libraries as distinct with respect to syncing iDevices.
If you have configured iTunes to organize your media and to copy files to the iTunes Media folder, then the complete library will be contained within (by default) C:\Users\username\Music\iTunes, with a structure of subfolders and key files like this:
If this is the case, you can (with iTunes not running on either machine):
For future reference, you can use turingtest2's user tip on Backup your iTunes for Windows library with SyncToy for a suitable method to backup your library to an external drive - you can then, with care, use this to keep the copies of your common library in sync across your laptops.