Why does my router have a power on/off switch?
It's largely for convenience.

Not having to crawl under desks to plug and unplug things you aren't using isn't exactly my favourite pastime, and I assume it wouldn't be yours either.

Adding a switch on the front panel to just simply turn a thing on and off without turning everything else off, nor having to dig into the mess of cables near your typical extension cord and wondering which thing to unplug just seems like a no-brainer. Also, this reduces the strain (cycling) on any of the connectors used, as @Tony Steward pointed out.

About routers, some people still prefer turning them off when they're not in use, mainly to conserve power. Uplugging the power jack to them may be an option too, but again a more painful one. Having a switch also makes it easy to power cycle bricked devices.