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A brown dwarf is more massive than a gas giant.

It is massive enough to fuse deuterium, but not sustain the process. Jupiter would be a brown dwarf if enough mass had been available, for it to be roughly 13x its current mass, but there simply wasn't nearly enough matter in our proto-solar system, the Sun "gobbled" most of it up.

A red dwarf, the lowest mass stars, are about 80x Jupiter's mass, and they can and do sustain Hydrogen fusion. They are the faintest stars that exist, and most of that faint light is emitted in the Infrared.

The difference really boils down to mass.

A gas giant such as Jupiter isn't massive enough to generate nuclear fusion in its core.
But an object with about 70 times Jupiter's mass (according to current theories) has enough mass to initiate deuterium fusion in its core. Its not full nuclear fusion of hydrogen to helium as in our sun, so the brown dwarf is only about as warm as an oven and it won't shine in visible light (but should be detectable through its infrared radiation).

Currently there is some debate as to what criterion to use to define the separation between a brown dwarf and a giant planet at very low brown dwarf masses (~13 MJ ), and whether brown dwarfs are required to have experienced fusion at some point in their history.

http://en.wikipedia.org/wiki/Brown_dwarf