The difference is one of observed behavior.
In phosphorescense, a material captures light (photons) from a source, and continues to emit light after the source is removed. The best example of this would be a glow in the dark toy. The toy glows after the light is turned off. It also glows when the light is on, but we generally can't see it.
During flourescence, the light emited is dependent on a source being present. When the source is removed, the material's emission of the captured photons immediately stops. Flourescence also tends to be highly specific in regards to the wavelength (color) of light captured and the wavelength of light emitted. These specific properties are exploited for the purposes of scientific imaging.