What does doing the sunnah have to do with loving the prophet? (Sunni View)

I don't understand what doing the sunnah has to do with loving Prophet Muhammad (peace be upon him). After all, we do our deeds for Allah's sake and not for Prophet Muhammad's sake (peace be upon him). From my understanding of Islam, a classic example of not loving the Prophet Muhammad (peace be upon him) is not following the Sunnah. However, it seems to contradict doing things for Allah's sake and this is confusing me. I just don't understand why I can't love the Prophet (peace be upon him) without doing the sunnah (if that's true).