Beryllium is very light and very rigid. These are technically desirable characteristics for something that has to vibrate up to 20,000 times per second (or more).
I suspect that the gold tweeter is just flash - the ribbon one offers the advantage of having low distortion and next to no ringing.
The three probably sound differently from one another, but without being able to compare, I couldn't say which is "best." My guess would be that the beryllium and the ribbon would outperform the gold one (assuming that the gold color is plating rather than anodizing). If the "gold" tweeter is just anodized, then it should perform to the ability of its underlying metal (aluminum probably rings more than beryllium).
Now the above are generalities, and specific implementations of aluminum, beryllium, or ribbon tweeters may perform better (or worse) than the tweeter material alone would lead one to expect. So everything I've said is mostly useless, but I'm trying to answer your question of "Why would a customer prefer one tweeter over another." The simple answer is "they probably sound different."
Last Edit: Nov 27, 2018 8:04:48 GMT -5 by boomzilla
The beryllium tweeter, in theory, will have a faster response time than the Audax(gold)tweeter due to it's decidedly lower mass. As to differences in sound quality an audition would be in order. The Lore Be, per the Tekton web site,costs $500 more than the Lore with the Audax tweeter,and that would be a factor to consider in terms of performance tradeoffs. Remember though, that the Audax is no slouch, and is found in many great speaker designs. Ribbon tweeters are inherently more directional than domes, and thus would create a narrower, more focused sweet spot. The ribbons in my Apogee Centaurs are a dipole design, and are not quite as directional horizontally. The vertical window of the Centaur ribbon is fairly narrow, however. . As Boomzilla states, ribbon tweeters have extremely low distortion, very fast response times, and are virtually free of ringing.
That Audax Gold tweeter in the original Lore is used in a lot of musician gear implementations, such that if you go see a lot of live music performances (electronic, not acoustic/classical), what you hear from the Lore will sound very similar. I have it and agree; it is a superb tweeter.