The single major difference between a star and a planet is that stars always produce radiant energy through nuclear fusion and, briefly at the end of its lifespan, nuclear fission; while planets may or may not produce energy on a much smaller scale through core nuclear fission and gravimetric force. It is this radiant energy that allows stars to be seen by their own light, while in the non-infrared bandwidths planets can only be seen by reflected starlight. Neutron stars continue to emit radiant energy, mostly in the gamma range, and thus still fall under an extreme category of star. Such objects as black holes which produce energy only as a gravitational byproduct may once have been stars, but are so no longer.
As a direct result of this different source of energy, stars continually create new and heavier elements through nuclear fusion. Planets only ever break down existing elements.
Size alone cannot determine the difference between star and planet. The smallest white and brown dwarf stars are smaller than the planet Jupiter, but are far more dense. The largest known stars are exponentially larger than the largest known planets.
Substance alone cannot determine the difference between star and planet. Gaseous planets may have similar proportions of hydrogen to active stars, but don't use that hydrogen for nuclear fusion. Thus, Jupiter is sometimes referred to as a star which failed to ignite.
Visually, stars often appear to "twinkle" when observed through a planetary atmosphere such as that of Earth. Planets have much more of a steady glow.