I've never used a time code clapper, so this is just a guess.
Back in the days when film was edited using a work print on a flatbed viewer, audio tracks were transferred from the 1/4" original tapes recorded on the set (using a reel-to-reel Nagra) onto magnetic film - called "mag-stripe" or "full coat" depending on whether you were shooting 35mm or 16mm. The Assistant Editor then would synchronize the rolls of pix and sound by watching the sticks drop. At the point where the top stick made contact with the slate, you would mark that frame of film and match it up with the frame of mag where the sound of the clap began. Everything after that would remain in sync until either the camera or the Nagra stopped rolling. This would be done for every single printed take of every single setup of every single scene in the entire movie -- literally thousands of takes for a feature.
As someone who has done this for a couple productions I can tell you that it is far from an exact science. Depending on how the 2nd AC held the slate in any given take, it could be a real challenge to try and determine on exactly which frame the sticks made contact; the slate may have been held far away from the camera, or the sticks may have blended in with the background, or the slate may not have been held steady. In addition, there often is background noise on the set that obscures the clap sound, sometimes even several similar bangs so you can't tell which is the one you need to use for sync. I wasted many an hour using trial and error when syncing work prints.
When digital audio tape recorders replaced the reel-to-reel Nagra, they generated a time code which could be transmitted to the slate. The slate is designed so that when the clap sticks touch, the time code display freezes at that exact frame. Since, as far as I know, there is no way to record time code onto 35mm film, the camera instead shoots the frozen number for a second or two until the slate is removed. When the film is digitized for editing, it can be synchronized with the digital audio simply by typing in that time code number and matching that frame of audio with the first frame in which the numbers freeze on the slate. Much, much faster, easier and more precise than the old method.
If my guess is wrong, I'm sure someone will correct me.
