Pilates and Yoga have a lot of similarities. They both use the breath to focus your mental concentration. They both use the breath to keep you innately focused on the present, like a form of active meditation.
The breathing patterns in both are intended to actually enhance the execution of the movement. Abdominal breathing in Yoga is used to calm the nervous system.
However, in Pilates you will practice diaphragmatic breathing (lateral breathing), which prepares you for active movement and emphasises strong abdominal contractions on the exhale, to build more support in your core muscles. This is why Pilates will give you a much harder abdominal workout.
A lot of the exercise movements in Pilates and poses in Yoga look alike. However, Pilates is more about specifically designed movements where Yoga often is about holding a pose.