Actually, I think active pickups are noisier than passive ones.
It all comes from the fact that active pickups are called active, because they have a built-in amplifier. Not a stack, of course, but a small one that increases the output of the pickup and allows for instance the boosting of desired frequency ranges. (If I recall correct, B.C.Rich offers some custom shop guitars with more knobs and switches than a regular shuttle cockpit.
)
A few years ago I went out to update the pickups on my Epiphone. I had decided to get EMG-81's, because Hetfield had them and he sounded good. (Ah, those were the good'ole Metallica days...) Anyway, I marched into a music store that I had been recommended and asked for new pickups. I had my axe along and enquired for the EMGs. The clerk asked to see my guitar and as I opened the case he exclaimed: "DON'T YOU FUCKING PUT EMG'S ON AN EPIPHONE!!!".
The moral of the story (as I picked up from the two-hour lecture on pickup selection that ensued) was that sustain and tone come most of all from the body and neck of the guitar. The better crafted the body, the better the tone. And when looking to improve ones tone, the first thing to ask should be: "Is my guitar worthy of active pickups?"
I walked out with passive Seymour Duncans and have been happy with them. (Of course my Epiphone is sooooo crappy that the pickups don't matter much.
)
So, passive pickups are a bit more forgiving when it comes to playing mistakes. The active pickups amplify it all - every little snap, hiss and tinkle. (And give three times the output of passive pickups.)
Well... I hope that my ramblings are of some help.