"I have a Pro Tools Mix system. I need to playback audio tracks to my client's picture on digibeta. This digibeta has 'video sync' in and 'word clock' in. I have a black burst generator I've been syncing to. What's the difference between using video sync and word clock to lock the audio to the video? Which hole should I plug the black burst generator in to?"

Black burst is video sync, so that's the proper place to connect that signal. Whether you should sync everything to black burst or word clock depends, but usually either will do.

With analog video, the resolution of any video system is only as good as its frame rate. Black burst is considered the most stable timing source for such a system, and when audio gear syncs with video black burst is used as the reference to which any synchronizers or digital audio equipment derive their timing. Of course some house sync generators provide black burst and word clock (and sometimes even super clock) so everything in the studio can easily reference the one source, which is preferred.

In a digital video system, which also has digital audio, you now have word clock available. Word clock has a higher resolution than black burst - thousands of samples per second versus 30 or so frames per second. Thus it is generally considered best to go ahead and lock to the word clock. In your case the word clock and the black burst are almost certainly going to be resolved to one another inside the digital Beta deck, but you may as well just use the word clock anyway. Besides, in order to use black burst with Pro Tools you need additional hardware. You can just tie the digital audio out of the Beta deck to a digital input on your Pro Tools interface and use that to clock the Pro Tools system. There are other ways to skin this cat, but this is the simplest.