The dimensions of any driver, cone or ribbon, influences its dispersion. (In the case of a round driver such as a cone or a dome, the length and width are the same. Often ribbon drivers are rectangular.)
In theory, smaller driver dimensions produce greater sound dispersion. Because drivers produce audio frequency vibrations, the wavelengths are large enough to be similar to the driver dimension. In general, a driver smaller than the wavelength produced tends to disperse the sound widely. As wavelengths get similar to or smaller than the driver, the sound dispersion becomes narrower. A dome tweeter has similar vertical and horizontal dispersion. A rectangular ribbon tweeter can have a more limited vertical dispersion if its much taller than it is wide.
In practice, there are also mechanical and manufacturing limits to size. But that tends to change as manufacturing methods develop.
The thickness of a ribbon, cone, or dome is a different question. It's a matter of weight. A thinner or lighter driver starts and stops moving faster in response to the electrical music signals. The very light weight of ribbon tweeters is probably the main reason why they sound different from dome tweeters. Weight is also one of the physical characteristics that determine a driver's inherent resonance frequency. As weight goes up, the resonance frequency goes down, allowing lower crossover points. But as weight goes up, the speed advantages are lost. So an engineer must decide how to balance these when designing a driver.
This page discusses some of this in greater detail:
RAAL: TECHNOLOGY
