Single coil are the "classic" , original Fender Jazz bass pickup and they tend to have a specific sound BUT, because they are single coil, they also tend to be much more susceptible to noise and interference - they have no inherent capability to filter out noise or other stray signals.
Dual coils, also known as humbuckers, are wound in a manner that the pickup itself can inherently filter out the interference and noise.
I understand that there are some humbuckers that do a very good job of replicating the original's tone.
I don't have a jazz bass and therefore have not messed with them, so can't advise on their specific sound differences, but Talkbass likely has a whole forum dedicated to the pros and cons of various PUPs.
There are also some folks that go to extreme lengths to try to shield and protect the originals from any kind of signal interference.