Why use halftone screen angles?

Go back in the history of halftoning and you'll discover that for multi-color presswork the halftone screens were always angled relative to one another. Why not just have all the screens at the same angle and be done with it? Well, when halftone dots are at the same frequency and angle then they will either print on top of each other (dot on dot)or they will print partially overlappingor they will print beside each other (dot off dot).
When dots print on top of dots you have wet ink sticking to wet ink (wet trapping) and white spaces between the dots. When dots print beside dots you have wet ink sticking to dry paper (dry trapping) with less white spaces between the dots. What does that mean? If you look at the images above you'll see that the dot on dot color looks darker and less vibrant than the dot off dot. The result is that the final blue hue, in this example, of dot on dot will also be different than the dot off dot blue because wet trapping inks reduces the ink's efficiency at filtering light. The white paper surrounding the dots also contaminates the perceived color by adding a graying effect and therefore the dot on dot printing will have less chroma (vibrancy) than the dot off dot.

The biggest issue though is that when there is slight misregistration on press the screen will shift from dot on dot to dot off dot causing the presswork to shift color and tone dramatically. However, by rotating the screens relative to one another, this wet trap/dry trap effect is randomized and therefore the color becomes more consistent when slight misregistration occurs.

The second major issue that occurs if all the screens in an AM/XM halftone have the same angle is that of moiré. Here, one of the colors has been slightly rotated - perhaps because of a small imaging problem, or because of a small press problem.
When this happens a very strong moiré appears when all colors have the same angle. However, by rotating the screens so that they are 30 degrees apart, there is some tolerance for small angle errors and moiré will not appear.

Using rotated screen angles for AM/XM halftones overcomes the dot on dot/dot off dot issues.
With FM/Stochastic screening, the same problems are overcome by using a different screen pattern for each of the process colors.