A robust design is one that not only is correct, but also ensures the functionality even when boundary conditions deviate from the ideal. It's a mix of stability, predictability and fault tolerance. Probably "reliable" can be used as a synonym.
At the same time, in all industries except CS "safety" has a specific meaning of not causing injuries to the user.
In the design of a drill, for example, if the motor is guaranteed to spin at the intended rpm independently of orientation, temperature and state of charge of the battery, that's a robust design. You'll hear the word "safe" only if it has two triggers to ensure both hands are on the handles during operation.