What is the main advantage of Switched-Capacitor filters?
A switched capacitor filter has a "building block" that is made from analogue switches and a capacitor. This building block is equivalent to a variable resistor whose value is modified by the clock frequency. You have to be aware of aliasing so you can't use the "building block" with signals that have frequencies that approach half clock rate.

A significant advantage is that the switching frequency can be altered "on the fly" and you then have a filter whose (for example) cut-off frequency can be swept from one point to another. Quite a useful feature and because capacitors are fairly easy to make on a silicon die they can be integrated too.

Being able to control the frequency allows you to alter two or more resistor values simultaneously and this means that the Q of the filter needn't be affected for instance on a sallen key filter that needs both resistors to remain in the same ratio to keep Q constant.

Given that the "frequency" of the clock can be digitally derived, you can use digital techniques to shift the clock's frequency and thus change analogue values.

A passive LC filter has a Q factor that is largely fixed by the component values. For instance a low pass RLC 2nd order filter might have a response like this: -



This has a cut-off frequency of 1.59 kHz and is a decent approximation to a maximally flat Butterworth response. If I lowered the capacitance from 1 uF to 0.1 uF you would see this altered response: -



The cut-off frequency has changed from 1.59 kHz to 5.03 kHz but the Q factor has gone from 0.707 to 2.236 and now you see a sizable peak in the response and if you look at the transient waveform for a step input, there is clearly more ringing.

Picture calculator source.

An active RC filter like a sallen key 2nd order filter is just as problematic in that (as previously mentioned), to keep the Q constant whilst changing the cut-off frequency, you need to adjust two resistor (or capacitor) values. This makes it difficult to control.