The following is information on how each console handles palettes and colors.
The NES has 56 available colors for sprites and backgrounds tiles. These colors are defined by hardware and cannot be changed with software. Colors are referenced using a two-byte hexadecimal value and a color table, sometimes referred to as the systemor native palette.
Above is an image of the NES system palette in a four by sixteen table. As you can see, the NES actually has sixty-four colors. Only fifty-six are unique, so it is commonly cited as only having fifty-six. (Some say only fifty-five, claiming color 1D is so close to 0D and friends on video screens it might as well be the same color.)
Assets can use up to four colors (with the first color being shared by all assets, sometimes called the "transparency" color—more on that later).
In summary:
SNES sprites are sixteen color and have access to eight subpalettes. Unlike the NES, there are no hardware limits for colors; the game designer can freely specify what colors are used in the RGB palette.