A complete color wheel includes purple, but a spectrum or rainbow does not include purple, only violet. There is no single wavelength of light that appears purple in isolation. Depending on how deeply you want to go into the physiology/psychology of color, you could think of purple as your "seam". If you have ever looked at CIE color space, it looks kind of like a tilted upside down U with a straight line along the bottom. The attached file is from the wikipedia page about CIE color space. The U shows the location of each single wavelength (blue numbers between 380 and 700 along the outside of the U), and the straight line is thrown in to depict purple. CIE color space is an experimentally derived color wheel, which is distorted away from a circle because biology is messy and because there are no single wavelengths that appear purple.
The fact that color circles are a reasonable way to describe the perception of a one-dimensional spectrum (plus purple) depends on the way that neurons in the visual system represent color, that is, in an opponent fashion. These neurons change their activity level in opposite directions to different wavelengths of light. For example, some neurons increase their output to red light and decrease their output to green light; some do the opposite (+green | - red). Blue and yellow make the other opponent pair (+blue | -yellow or +yellow | -blue). Complementary colors are due to this opponent processing, and the exact colors you see depends on the relative response of these 4 different opponent processes. Color vision is complicated, but scientists have been working on it for hundreds of years so there is a lot known about how it works.