I used to use Spectra when flying large stunt kites. The big advantage in that application over nylon or dacron is that it has very little stretch so the commands to the kite are much more precise and it gives a much more exact feel when flying the kite. The other advantage is that with kite flying it is common for the two control lines to be crossed or wrapped several times for most of the flight so they rub against each other and extra abrasion resistance is a good thing.
In a cave however, I'd avoid it like the plauge. If you snag yourself on a braided nylon line, one of the indications will be the slow increase in tension as you begin to stretch the line - becuase it stretches. That stretch also ensures that the energy from any diver contact or pull is dissipated slowly over the lenght of the line reducing the tensile strenght required compared to a line with very little stretch (lines with little stretch have to be stronger to avoid breaking).
Plus repairs are more easily made with nylon as you can stretch the line slightly to re-tie the broken ends of a line rather than having to splice a piece in on a line that does nto stretch. And, if you spliced a piece of line into a spectra line, the very limited stretch would make it more difficult to ensure the line was tight after the repair was finished.
I am not sure if spectra sinks or not, but bug is correct that one of the reasons nylon is used is that it sinks.
"Stronger" is nice except that more strenght is not really needed in cave diving and smaller line diameter does not offer any particular advantage as below certain point the ability to increase the lenght on a spool is offset by poorer handing traits - you want to avoid tangling your line as much as possible and on balance braided nylon line handles quite well.
Finally, I can pick up 500' of braided #18 or #24 surveyor line for $4 or $5 at the local hardware store and I can get a 1200' or so of #24 in a dive shop for around $25-$30, so when the nylon line on a spool gets worn it is cheap to replace. If divers are paying 10 times that much for line, they are going to be a lot more hestitant to change it whenever it finally needs changed.