Stars twinkle because of the Earth's atmosphere. Stars are so far away that they appear as point sources of light so any disturbance of the light by the Earth's atmosphere is easily visible.
Planets are closer and appear more as a disk than a point of light.
Any disturbance is not so visible because it is 'hidden' by the rest of the disk. If the central part of the image is distorted as it passes through the atmosphere, chances are you won't see the distortion because it will not reach the edge of the image. The proper name for twinkling is scintillation and can be a real problem for astronomers using telescopes where the affect of the Earth's atmosphere is greatly enhanced by the magnification. That's why most observatories are high on mountains so the light they look at has to pass through less atmosphere.