The direction of a comet's tail has nothing to do with the motion of the comet. Contrary to expectations the tail streams away from the Sun. Dust from the comet is blown by the pressure of light from the Sun and produces the main tail that then streams away from the Sun.
Image showing how a comet's tail always faces away from the Sun. Note also how the tail gets longer and brighter as it gets closer to the Sun.
Sometimes a second tail is visible. The ions or charged particles associated with the comet interact with the wind of charged particles that the Sun emits producing another less visible tail along the direction of the solar wind that spirals out from the Sun: the ion tail.