there are many diffrerent types of active and passive pickups, it is very hard to generalize the pros and cons, just try whichever pickup you want and see how they sound
theres one con for active pickups, you have to replace the battery
True on the replacing the battery... but it is rated for hours and hours of play, here is some info from the EMG site:
How long does the battery last?
All EMG pickups and EQ systems are designed for extremely low current drain. In addition, the pickup jack included with all models has a switch that disconnects the battery when the guitar is not plugged in. To maximize battery life, you should always unplug your guitar when it's not in use.
The Specs Page includes current requirements and estimated battery life for each model. Generally, each pickup requires about 80 microamps (uA), except for the Vintage Series pickups which require 220 uA each. EQ circuit requirements vary widely but are higher than pickups.
For your reference, a standard 9 volt alkaline battery provides 580 milliAmpHours (mAh) of power. That means that it will provide 580 milliAmps for 1 hour or 1 milliamp for 580 hours. There are 1000 uA per mA. You can figure the approximate battery life of any setup by adding up the individual power requirements, then dividing 580,000 by this total. Here's an example:
1 - EMG-81 = 80 uA
2 - EMG-SV = 220 uA*2 = 440 uA
1 - EXG = 410 uA
Total required = 930 uA
Total life = 580,000 uAh/
930 uA = 623 hours
If you left your guitar plugged in day and night, the battery should still last a month. Under normal playing conditions, you would probably be looking at changing the battery twice a year. Of course, you should treat these numbers conservatively and not try to drain every last uA out!
Also if your thinking about getting passive pickups and putting them in your guitar, keep inmind you'll need space for the battery, yes, that means routing out your precious guitar!! There are some great passive pickups out there if you are not willing to do that.