There is one main difference between a star and a planet: A star is 'burning' fuel to generate it's own light while a planet is not burning fuel and is only visible due to reflected light. This is, of course, a general rule and there are exceptions.
The main ones being 'dead' stars, black holes, and neutron stars.
We should first consider planets as they are quickly dispensed with in this context. There is generally little heat generated by a planet and it normally comes complete with either a solid or molten core beneath a solid surface. In this context we are treating solid to mean not emitting internally generated energy caused solely by self combustion. The planet Earth for example generates internal heat as evidenced by hot springs and volcanoes. That heat, however, is the result of pressure caused be gravimetric force, gravity is squeezing the surface toward the center and that pressure creates enough head to melt rock. What ever liquid gases found on the surface of a planet are strictly a result of the atmospheric temperature being high enough to melt the liquid. That surface temperature resulting from one of both possible sources which are the heat generated upward by the molten core of the planet or be external radiation from a nearby star.
And all that is a complex, scientific way of saying a planet has a solid surface and is not burning itself up.
A star on the other hand is burning itself up. It shares the property of having a molten core the the same reasons as a planet, that gravity is such that it pulls the heavier elements together to form a semi-fluid core. The 'surface', however, is not solid as it is a ball or light gaseous elements (predominantly hydrogen and helium) that are changing composition (burning) from a given element to another element - most commonly from hydrogen to helium and from helium to... well, you get the picture.
Each elemental change gives off energy if the form of radiation including visible light, gamma rays and x-rays etc. As the elements resulting from this nuclear combustion climb the atomic number scale and become heavier they come more susceptible to the pull or gravity and are drawn to the core. Thus completing that cycle.
A 'dead' star is one in which the cycle has totally exhausted the hydrogen fuel initially available to it. It could technically be called a planet in that regards, I suppose, but it comes down to a matter of size. It is simply too big to be called a planet since it still is the size of a small star. This is the eventual fate of all the stars in the galaxy that are not large enough to be pulled into a neutron star or become a 'black hole.'
The neutron star is one in which the gravitation force generated by its collapse is sufficient to reduce the mass of a star into a ball roughly 12 kilometers or so across with a surface layer of neutronium and a gravity at the surface to be measured in thousands of G's. A 'black hole' was of sufficient size to go one step further and collapse into a 'singularity' with a surface gravity of such intensity that not even light can escape it. It would be totally undetectable except for the radiations given off as normal matter that has been sucked in passes the 'event horizon', which really is another article entirely.
One other small note before I leave you. Given its composition the planet Jupiter just missed being large enough to ignite and become a second small star in our own system. Now if Saturn or one of the other two gas giants were moved to Jupiter, I wonder...