Why does drilling underwater hurt me so much?

Being underwater crushes your digger.

You'll note that, on aquatic planets, your temperature gauge is replaced with a pressure gauge, that works a bit differently. The pressure gauge goes up as you swim deeper underwater, like the environmental temperature does, but it does not go up as you drill. In other words, there's no "cool down" period for your drill like there is on terrestrial planets.

Instead, the higher the pressure level, the more damage you take when using your drill. This means you can drill non-stop underwater for as long as your health can take it (and you have oxygen), but you'll lose health steadily the entire time.

The distinction here is hinted at in the in-game help documentation (as in this screenshot of the "Depth" help entry):

The same plating upgrades that help you cope with environmental damage on terrestrial planets will also help reduce the pressure damage on water planets.