Why does my sleep score not match how I feel?

Most scores compare you with general averages and look only at the night. How rested you feel depends more on how the night compares with your own normal and on the day before.

Can a smartwatch tell if I slept badly?

It can estimate it. Watch data plus a few personal details picked out awful nights from excellent ones most of the time, but was less reliable between ordinary nights, so it suits spotting a bad patch rather than judging one morning.