I've always liked them because of the artistry involved. Not that the artisanship of briars can't be impressive, but to me, the meerschaum with its ever-changing colors, even in the classic shapes has always been beautiful. That's purely personal aesthetic.
Because of the way the material in Meerschaum conducts heat and moisture, it makes the smoke cool and dry and since the Meerschaum doesn't impart its own taste to the tobacco, it will bring out different flavors than ina briar. As others have said, it is a little more resiliant for smoking bowls back to back and developing a cake is not necessary.
There is a downside to Meerschaum. Even in the good ones the drilling can be a little wonky. There are varying qualities of block, as some mentioned. If you drop it on the concrete...Well, goodbye. :crying:
All I really have to say is smoke one. If it's for you, you will know right away.