The stationary case is different in kind, not degree, and that is why it is much worse.
With the car stopped and the engine running, the input shaft is turning with the engine and the output shaft is stationary because the wheels are not moving. There is no speed you can select that matches both — the difference is the entire engine speed.
So attempting to force it into gear means driving gear teeth into other teeth with a large speed difference and no way to bridge it. That is the grinding noise, and it is teeth being damaged rather than a friction surface being worn.
This is also why the clutch pedal is required to select first gear from rest, and why a car with a failed clutch cable can sometimes be driven in the higher gears and cannot be got moving from a standstill.