"Brightness" is merely a function of the individual tweeter and crossover design. The material used to manufacture the diaphragm is irrelevant.
There is definitely a difference, both in theory and practicality. In order to vibrate quickly enough to reproduce the top octaves, designers must choose a material that has a very high stiffness value, and very low mass. The stiffness of the material allows the tweeter to reproduce the highest octaves cleanly, without distortion, or dome breakup. A low mass is required for proper high frequency extension (the tweeter needs to vibrate pretty fast to play at 20khz), and transients.
Metal domes (aluminum, titanium, etc.) have a very high stiffness/mass ratio and exhibit the best pis tonic movement of any material. However, metal, as you may know, also suffers from a complete lack of self-damping. Imagine a bell. After you''ve struck its metal sides, it continues to ring for seconds beyond the time of impact. On the other hand, stamp on your carpeted floor. You''d hear a thump at the moment of impact, but no drawn-out sound. The same can be said of metal tweeters. Although they have a high stiffness/mass ratio, they lack internal damping and suffer from "ringing" which can be seen in a simple impulse response curve. Designers, wanting to take advantage of metals'' high stiffness/mass ratio, have come up with a number of ways to deal with a metal''s lack of internal damping, with varying degrees of success. One way is to apply damping material to the metal''s surface, but that of course, raises the mass of the tweeter and consequently lowers the tweeter''s stiffness/mass ratio. Other designers have sought their own creative ways to tune the tweeter and push the ringing into the ultrasonic frequencies above 20khz, again with varying degrees of success. This ringing will still be detectable by an impulse response, but these designers claim that the ringing is ultrasonic and inaudible to the human ear.
Another option is silk fabric, which does have significant internal damping but does not have as high a stiffness/mass ratio as metals do. An impulse response curve of properly designed silk tweeter will usually show no ringing.
Qualitatively, a metal dome tweeter will sound very detailed (while critics of metal domes say they usually suffer from being bright.) Silk domes, on the other hand, are said to have more of a "smooth" sound.