Ribbon tweeters seem to be a noticeable improvement over most if not all dome tweeters. To me, they have a cleaner sound overall. Even the best dome tweeters can impart a sibilant ssss sound to some recorded music, but the ribbons I've heard don't. This difference can be subtle.
However, good ribbon tweeters, such as the RAALs, are expensive.
Ribbons are said to produce high levels of distortion at low frequencies (depending on which model ribbon tweet). Therefore just how they are integrated into a loudspeaker becomes important; the crossover frequency and slope, if chosen carefully, can avoid this problem.
They produce sound up to very high frequencies, well above 20 kHz, but that probably only matters to your dog. The best explanation of why ribbon tweets sound better is that their one moving part, the aluminum ribbon (which both carries the AC signal and moves air) is significantly lighter than the voice coil plus dome in a traditional tweeter. The ribbon responds much faster to a signal.