Pilates breath, also called Lateral breath, differs from Pranayama slightly. Instead of always breathing in and out through the nose (as is taught with Pranayama) Lateral breath encourages exhaling from the mouth.
In my personal experience, this helps with fluidity in the aerobic Pilates exercises. Breathing in through the nose can also help warm the body faster than if one breathed through the mouth on the inhale. In both Yoga and Pilates, keeping the body warm is key to success- the muscles will feel more relaxed and elongated with warmth.
There is no "wrong" way to breathe when exercising, but there are ways to improve your performance through modified breath. Please note that you won't be singled out if you're breathing differently than your neighbor. When I instruct a class, I just want to hear my students breathing; it's nothing to be shy about! We all have to breathe!