FFLs are required to keep 4473s indefinitely (edit: 20 years - thanks for the update OhShoot). The 4473 has the firearm serial number on it. When they close the shop, the forms go to the BATFE.
The biggest difference in the 4473/background check and registration is that with registration the government could easily determine all the guns you own - probably with a simple database query.
With the current model, if a gun is involved in a crime (and recovered), the government must go to the manufacturer to see which distributor received it, then go to the distributor to see which dealer received it, and then go to the dealer to see who purchased it. Not only that, but the dealer can keep their 4473s filed in a number of ways (including chronologically). It a laborious process to trace a gun.