A twin needle is useful for when you need to sew stretchable topstitching (like when hemming a knit). You can also use it for a decorative effect.
While a coverlock machine is purpose-built to produce a stretchable topstitch, not everybody has such a machine. A twin needle uses two upper threads and a single under thread. This causes a zig-zag effect on the back, making the stitch stretchable. Meanwhile, you get two parallel lines of stitching at the front.
Animated GIF by Ashley