This is a low yield topic but let me tell you what I understand about it.
The Medulla provides the inherent rhythm which means the drive for respiration even without thinking about it. This is actually the respiratory center and if you have lesion in it you'll have central respiratory failure.
The Pons controls the pattern (like how long is the inspiration and how long is the expiration and the transitioning in between them). A lesion may not necessarily cause respiratory failure it will cause an abnormal pattern of breathing. You will lose the fine tuning but the main drive is still there in the medulla.
The pons has to centers:
Apneustic center, if this activated then you'll have prolonged deep inspirations with short expiratory phases but this is always under inhibition by the above pneumotaxic center.