The difference is a magical property of matter called transparency. When carbon sits in a random alignment, you see black coal, but if you crystallize the carbon, you get a transparent diamond. the difference is the atoms alignment to the matrix of reality.
With opaque inks, such as carbon black and titanium dioxide, a photon hits the atom and depending in the pigment, either be absorbed (black) or reflected (white). White light is actually composed a many different colors of light. For a colored pigments such as red, any none red light is absorbed, while all of the red light is reflected.
For transparent inks, the light is either absorbed or transmitted. Clear allows all light to pass, where translucent red allows red light to pass.
Translucent inks need something else to reflect the light. This can either be a white or light color shirt or an underbase. Process color was invented to allow a full spectrum of color to be created, which can be separated into film with a compute. In the old days, separations was done use colored filters.
Pigment inks directly reflect the light. Given an ample volume of pigment, pigment ink can be printed on black.
Now we get to semi-opaque. These inks are a compromise between transparency and opacity. Where a process color requires 4 screens to to produce a single hue, an ink mixing system can mix a single specific color semi opaque ink of a specific hue. The pigment density of these inks is not high enough to be truly opaque, so many times they need an underbase of white. By halftone the white, the darkness of lightness of the hue can be varied. This translates into a "race shirt" where the inks is produced with the team colors, then have the shadow detail of the car expressed in the white underbase.
The semi-opaque ink is also not limited to printing on white. When one color is printed on another color, a third color is created, but the top color will take priority, but exactly how much will depend on the specific opacity of the ink. This is where color separation software comes in. If a computer knows the opacity of the inks and is told how may inks can be used, the software can decide the best mix of colors and what order they should print. Reducing from full color to simulated process is not an exact science, so skill and experience can vary the quality of the results.
