Stanley, there really isn't a very clear definition and this is still a keenly argued point.
Browns dwarfs burn deuterium. In models this happens if they are more massive than about 13 times Jupiter. The weakness of this that we think isolated brown dwarfs could condense from a gas cloud that are less massive than this; and young brown dwarfs won't have got around to fusing deuterium.
Planets must form from the disk around a star. This is ok, but: brown dwarfs may also form from the disk and it is also possible for planets to be tidally stripped from their stars and be found alone in space.
Planets must have a rocky core. This used to be thought definite, but now we think maybe sometimes planets can collapse from a gas instability in the disk in some circumstances, without the need for a rocky/icy core. It is true that brown dwarfs should not have a rocky core. However as an observational definition this is fairly hopeless since we can't even tell yet if Jupiter has a rocky core.
A flavour of the controversy can be gleaned from reading between the lines of the IAU statement on the definition of planets vs brown dwarfs.