Purple has a lot more red. (157, 0, 255) vs. (128, 0, 255). Good to have learned something today...
I think using violet as a name for the entire color-range around (~128, 0, 255) is also common. So in a sense purple is an element of the violet color-range. But as points they are distinct. I think purple is more specific - as a color-range it'd cover less area.
I think of purple as encompassing both indigo (blueish purple) and violet (pinkish purple).