The advantages are no moving parts (therefore more resilient), lower power consumption (in some cases) and faster boot up times for the OS and programs. The downside? They are very expensive!
I've just come across this by Sans Digital. No idea how well it'll work, but its an interesting idea for anyone who wants to replace the hard drive in their laptop. Its basically an adapter which can take two Compact Flash (CF) cards, and turn them into a 2.5" SATA hard drive, all for $99.00.