On #BSD and #Linux (maybe only before systemd)

shutdown -r reboots the computer while allowing processes to cleanly stop.

reboot reboots the computer without waiting.

Are there any times where you'd want to reboot rather than shutdown -r?

As far as I'm aware reboot uses shutdown with the proper arguments anyways unless you do 'reboot -f', at which point it sends the system call to reboot, instead of essentially being shutdown -r.