What are the the advantages of a switched capacitor filter over a digital filter?
The primary advantage of switched capacitor filters is that they can be easily implemented on an integrated circuit. You can get performance similar to an analog RC op-amp based filter using a switched capacitor topology, while avoiding the need for an ADC, DSP, and DAC on a chip. 

Switched capacitor circuits use capacitors and switches to emulate the behavior of resistors. Additionally, the frequency response is determined by the ratio of the capacitors, so even low frequency filters can be easily realized on-chip. The real benefit for IC implementations is that while the absolute value of capacitances and resistances have a poor tolerance, the matching between similar devices is very good. This makes it possible to implement relatively high precision analog filters on a chip.

In an integrated circuit, you would choose a switched capacitor filter for the following reasons:


Minimizing chip area is a priority
You will not be doing significant digital processing on the chip
The output of the DSP would be an analog signal


Practically, you would not use a discrete switched capacitor filter (using op-amps, capacitors, and analog switches) at the board level - you would use an active RC continuous-time filter. There are switched capacitor chips that can provide good filtering results with just a few additional components. Using a general-purpose DSP on a board level design will require additional programming, and may not have analog outputs.