In my personal conception, violet is the kind of colour at the lower edge of the rainbow, which is a single wavelength. And purple is what the brain constructs. However, of course, the names of the colours are themselves vague.
And as such, they are both a construct of the brain, as any other colors, like... white.
What we label as "violet wavelength" is only a narrow projection of our experience outward. Case in point, we don't have such colorful (eh) names for other EM wavelength.
I say narrow because you could take this pure laser and change th surrounding and you will inevitably perceive it differently, even though the power and wavelength are the same.
"Purple" is a mixture of red and blue.