It has one purpose: To make the video quality better, but primarily for video that has to be "squeezed" a lot in order to make it fit. If the total length of video on your DVD is going to be 90 minutes or less, then encode at 6,500,000 average bits per second and select single pass. To fit more video on a single DVD, you must use a lower bitrate. Fewer bits per frame will result in more "blockiness." However, the encoder can use fewer bits for scenes with little motion, and then use the bits "saved" to encode with more bits for fast moving motion. This is what "variable bitrate" encoding does. To make it work really well, the encoder has to look at every frame from the video to know which are moving fast and which aren't. At high bitrates, little can be gained, but the lower the bitrate, the more important two-pass becomes. By the time you get down to 5,000,000 bits per second, two-pass is almost essential if you want good looking video.

There is no difference in stability, compatibility, or anything else.