The "spectrum" of light refers to the relative power at each wavelength in the range of wavelengths of concern, in this case the visible wavelengths.
"Color" refers to the way we perceive various mixtures of wavelengths of light (that is, how we perceive the spectrum of light). Most human beings have three types of light receptors (rods) in their retinas that respond rather broadly to three different overlapping ranges of wavelength (Short, Medium and Long) centered around 440 nm, 540 nm, and 575 nm. Our brains receive neural impulses from the rods and interpret them to provide a perception of color.
The important point is that with only three types of receptors, our eyes are not able to distinguish the whole spectrum. Our brains really just report the ratio of stimulation of the three different types.
OK, that's all background info. I think the core of your question boils down to "Is there a way we can use our eyes to really perceive the spectrum of light?" If there are no constraints, the answer is "yes". For example, you can look at a light source through a prism or a diffraction grating, and it will let you see the whole spectrum. Each point of light will be spread out into a rainbow line, where each position along the line corresponds to a different wavelength of light. There are other, more complicated ways, too. If you would like to experiment with a diffraction grating, you can get one from, e.g., Edmund Optics.