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Unless you have noise…
Unless you have noise, you may not even think about filters, let alone whether you should use an active filter or a passive one.  Let’s start out with a definition.  What is the main difference between the two types of filters?  Simply, an active filter uses an active component, one that is connected to a power supply.  A passive filter only uses the power that is supplied by the signal it is filtering. One other difference is that active filters never use inductors. An active component is most often an Op Amp, but it could just as easily be a transistor.
I’ll use the passive, it has less components
In figure 1 we see two low pass filters.  Figure 1a is a passive filter and figure 1b is an active filter.  They both have the same cutoff frequency and the same gain.  But that is where the similarities end.

Figure 1: 1a Passive low pass filter, 1b Active low pass filter
I said that they have the same gain, but I made an assumption, that R1 = R2 in figure 1b.  In that case the gain is R2/R1.  One big difference between the two filters is that the gain in 1a is fixed, while the gain in 1b can be varied by changing the ratio of R2:R1.
A second differnce is the output impedance.  The output impedance of the active filter is low because the Op Amp is driving the output.  Not only is the output impedance low in an active filter, but the Op Amp acts like a buffer, decoupling the inputs and outputs.  For example, both circuit will work similarly if a high impedance load is connected to Vout, but if a low impedance load is connected, the circuitry connected to the Vin side of the passive filter will be affected by the increased load placed on the circuit.
As for cost and complexity, the passive filter wins, hands down in many situations.  If you can get away with using a passive filter, you should, but if you need the advantages provided by the active filter, the additional cost and added space is well worth it.

How do we choose?
There are three situation where a passive filter shines.  The first is when the signal you are filtering is a power circuit.  That is, you actually provide power from a circuit, but it is noisy and needs to be conditioned.  Removing ripple from a DC signal is a good example of this type of circuit.
The second, is when you need a simple circuit with just a little bit of filtering.  A simple RC will give 20 dB/decade of noise reduction.  This is part of the KISS principle, keep it simple stupid.  Don’t make things needlessly complex or costly.
The third situation where you would want to use use passives is for high frequencies.  That is when f > 10MHz.  You want to use an Op Amp capable of frequecncies about 100 times higher than the frequency you are filtering.  So if you have a frequency of 10 MHz that means you need a 1GHz Op Amp.  The choices are few, and the cost is high.  This is a situation when you will be best served by passive components.

The amazing active filter
The active filter has more componenets, requires a connection to a power source and usually costs more.  That may sound like it is a bad choice but it has some amazing properties that make life better.  As stated above you can control the gain of an active filter, set it where you like (for the most part).  Another great attribute of an acitive filter was mentioned above, it isolates the input and output signals.
In low frequency casses, inductors tend to be large and costly.  An active filter, which does not use inductors, can filter a signal for less money and using a smaller footprint.  In fact, in some cases the active filter can be mineaturized where the passive will never be anything but large and heavy.
It should be mentioned that it is usually easier to design and tune an active filter than a passiver one.

Other choices
An active filter uses an active element such as an Op Amp or a transistor.  Recently a new active element has come on the stage, the switched capacitor.  While switched caps have been around for a long time, having switched capacitors available in monolithic form is new.  The typical accuracy of the cutoff frequency of a switched cap filter is ±0.2%, set by an external clock frequency.  They are able to change the cutoff frequency over a wide range simply by changing the clock frequency.  They also have a low sensitivity to changes in temperature.  The ability to change the tuned frequency on the fly is a big game changer.  You may not need this capability for every circuit you make, but for the times you do, switched capacitor filters are the bees knees.
Final thoughts
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