Mellophone has the same partials and fingerings as a trumpet.
French horn partials are MUCH closer. Starting on a C below the staff, the open partials are: C, E, G, really Sharp A or Flat Bb, C, D, E, F#, G, and then it goes nuts and its like all half steps.
Also, most people play double horns, which have the normal horn in F, and then a trigger which switches to the Bb side of the horn.This changes the sound slightly and the partials are more like a trumpets.
Then there's the whole right hand thing with stopping (which I suck at), tuning, and changing your sound with your right hand.
Also, most people use a trumpet-like mouthpiece on mellophones, which are cup-shaped, while french horn mouthpieces are conical.