Why does ice piece sticks on hand but not wood piece?
Good question ! The answer is that heat from your hand easily melts a bit of ice and the cold ice re-freezes that bit.
For the transition from wood to ice:
the wood is generally also a bit colder
the heat resistance is bigger:
the wood doesn't deform, so the contact area is small and
wood is a bad conductor for heat
It is not true that ice doesn't stick to wood at all: when the ice thaws a bit and then refreezes it can become quite well attached.
@BvU 's answer is correct.
It reminds me of the case when a child at my front door in winter touched a metal rail with his tongue. The tongue stuck to the rail which was very painful. But the child's clothes did not stick for the reason that @BvU said.
Tongue sticks because it was wet. The saliva freezes and acts as a bond. If your hand is dry, it probably won't stick to the ice.
That's part of what @BvU said in #2, but not the whole story. Ice on the cold object supplies the water. It is heat from the touching object that can melt some of that ice, only to have it refreeze a moment later that causes the problem. Flesh is warm and is a good heat conductor. The exterior surface of a glove is cold and a good insulator.
Separate names with a comma.