Stain is a colorant that is dissolved in a solvent, which means that it will not seep into the pores of the wood and can be removed more easily than paint. Stain also deteriorates on its own more quickly than paint, though.
You may also find additional options with stain, including some that highlight the grain of your wood rather than covering up. If this is something you are interested in, it is one noticeable advantage over paint, which is far more likely to cover up your grain to an extent. There are also stains that will cover up the grain of your wood as well, so you need to make sure you make the right selection and consider the possibilities.
With many stains you will also need to apply a finish or varnish over the top in order to protect the stain from deterioration and provide a glossier finish and look.