After the Restore, it is not set up to sync automatically. That's why "syncing" does not sync anything. You can set up automatic syncing, or load it manually by dragging songs to the iPod in iTunes.
To set up automatic syncing, select the iPod in iTunes (click device button on horizontal bar) to show the iPod's settings screen in iTunes window. Along left side (in the sidebar), under Settings, click Music. To the right, the iPod's Music settings screen is shown, where you tell iTunes how to sync songs to iPod.
At the top, checkmark the box for Sync Music. This turns ON automatic syncing. Below that, if your iTunes music library fits on iPod, choose the option to sync Entire music library. Otherwise, choose the option to sync Selected playlists, artists, albums, and genres. And make your selection from the lists below. Then, click Apply. Going forward, iTunes keeps the iPod "in sync" based on how you set up this screen.
However, if the click wheel still does not work after the Restore, that's probably a hardware problem.