What benefit is there in synchronizing my switching supply to the system clock

If your switching power supply frequency is close to your system clock but not exact, you could get mixing (from nonlinearities), perhaps causing interference in your signal band in analog circuitry. Usually it's the difference ("beat") frequency, not the sum, that can come to haunt you.

Mixing is a nonlinear operation (multiplying) and the sum and difference signals derive directly from the trig identity:

If they're synchronized, usually such problems will be minimized (perhaps a DC offset at worst).

For completeness I would like to note that switching regulators that exhibit a SYNC terminal usually have that capability for another reason. While synchronizing to the MCU clock can be advantageous as described above, this is very rarely needed and is thus seldom done.

The real reason regulators can be synchronized is for the case when you have multiple regulators in your design. Then, by feeding all of them the same clock frequency but with different phase, you can make sure they are not switching at the same time, which reduces input voltage and current ripple, eases input filtering requirements, and reduces EMI phenomena.

Another common reason is simply to make sure that all your regulators are switching at the same frequency, even if in-phase. This in turn allows you to focus your filtering efforts at a much narrower frequency band than if everything did its own thing at a who-knows-what frequency.