Lighter primers, as other people have said, make it easier to see details, but there are ways around that that I'll mention later.
Other than that, the big difference between white and black primers is what it does with the shadows and colors. You painted some red over white, and as you can see, the white affected the red. Imagine that same red painted over black. It isn't a strong effect, but white primer tends to result in more intense colors, while black results in more subtle colors. You can use that to your advantage. If you're painting a horde of 90s grunge zombies, priming black can keep them from being too colorful. If you're painting a court jester, on the other hand, priming white can intensify the colors.
With shadows, black and white shadows are painted very differently. With white, you (generally) paint the base colors, add some shadows by hand, then paint up the highlights. This results in subtle shading and colored shadows. With black, you generally put on the base coat, then focus more on the highlights, letting the natural darkness of the primer take care of most of the shadows. This results in darker, more intense shadows. Again, one isn't bettr than the other, but they do give different results.
Another thing to take into account is what colors your finished piece will have. If you're painting a bunch of black-robed cultists, prime black. If you're painting a bunch of white-robed clerics, prime white. You don't want to be fighting against the primer! Priming black can also work for lots of medieval armored figures if you're going for a quick-and-dirty paintjob. Primre them black, drybrush silver.
Gray is a balance between the two. You paint your shadows like with white, and your colors have more intensity than black, but it isn't as strong of an effect. In my personal experience, when I have primed gray, the difference between that and white has been minimal. It's more of a personal preference than anything.
Anyway, the details with black. Some people like black for their shadows, but want to be able to see the details. I've read abotu a couple of techniques for this. One is to take the black primed figure and give it a fast, very light drybrushng with white to pick out the details before painting. It gives you the benefits of black without the headaches.
The second technique is called zenithal priming (part of zenithal highighting), and is done to get very intense highlights and shadows. Prime the model black as normal, then go back and prime it a second time in white, but only from directly above. You end up with a miniature that is black with bright white on all the upper surfaces. The idea here is that the white primer coming down from zero degrees lands exactly where the light would. You then give it a base coat with a semi-transparent paint or a wash, and all of the shading is already done, just like that.
Some images I found demonstrating this. The first image is just the zenithal priming. The second image is with just a wash: