As far as I know, the principle advantage is excellent dispersion. This produces a very even sound field in the room and will result in a more natural and balanced mid range compared to almost all other drivers.
What you actually hear is the direct and reverberant sound field of a speaker and NOT the anechoic response seen in test measurements. Apparently, the key to natural sound is to excite the reverberant field in the same way as the direct sound. Most drivers have bumps and dips in their off axis response that do not follow their direct response and this is what creates so many room placement and listening position issues (and the dead giveaway of "reproduced" rather than natural sound).
The disadvantage is that large 3 inch domes are difficult to build (costly) and hard to integrate with other drivers; the result is that very few designers use them...from this perspective they are worse then normal cones which can be so cheap(a tenth of the cost) and therefore allow for a speaker with a better price point (or more profit for the manufacturer). Soft dome tweeters, of course, are the most popular design on the market because they offer similar advantages in the treble without the difficult costly construction (Xmax is tiny on a tweeter).
Here is some more technical stuff
Dome MidGiven the high cost/integration issues of dome mid ranges, I would say that they are only better if you go to very high end speakers and the proof is that very few manufacturers have had real success with this type of design....basically a poorly constructed dome mid (expensive) will sound a lot worse than a poorly constructed conventional cone(cheap).