is it the same thing as guitars? would a single coil pick up be much cleaner then a humbucker and a humbucker be better at distortion then a single coil.
Well, the distortion point doesn't really apply as single coils on bass don't sound as thin as on guitar.

Humbuckers obviously buck hum, but generally have a very distinctive sound to them with a pronounced midrange.

You need to try both on otherwise similar basses to get a sense of the difference in tone. There are also semi humbucking (precision bass) pickups on bass, which again sound different.

I use single coils and run my signal with overdrive most of the time.
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The best way to get an idea of the differences is to hear some players that play with single coil basses such as the Fender Jazz and compare their sound to players who play basses with buckers such as Ernie Balls.
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