The conception of the awakened Buddha is maybe a glimpse of our Buddha nature, or an intuition of it, or even just an interest in it. Through the gestation period, there may be more and more glimpses of it, or explorations of it through headless experiments, the Big Mind process, meditation and other practices. And the birth of the Buddha is Buddha Mind awakening to itself, as a field of awake emptiness and form, absent of any separate self anywhere.
(It is usually not as clean cut as this, but it may be a useful generalization.)
The Buddha growing up
The Buddha is born, and may realize its own nature clearly, but it still needs to grow up and mature in its expression in the world. Its vehicle in the world - this human self - has to reorganize and relearn how to function in the world and live its life within this new context of realized selflessness.
The first thing that happens is that it realizes that it has awakened to itself, yet also not. In the world and the lives of individuals, there are many examples of Buddha Mind not having awakened to itself, and of Buddha Mind experiencing confusion and suffering. So compassion arises naturally, and a desire to help - both with alleviating the suffering itself and in removing the causes of the suffering (if, when and to the extent they seek and want the help.)
Refining its instrument in the world
To do this, the vehicle for Buddha Mind in the world - this human individual - needs to be refined. It needs to continue to heal, mature, develop, and learn skillful means.
Deepening into the fullness of this human self
The more this human individual matures and deepens into the evolving fullness of what it is, the more it can connect with others where they are. It recognizes in itself what it meets in others. It becomes more deeply and thoroughly human, without having to defend or attach to any particular identities. It can allow the evolving wholeness of itself, with all the weaknesses and imperfections that goes along with being human.
Playing the game
Maybe most of all, Buddha Mind awakened to itself has to play the game. It has to take the experiences of Buddha Mind, when it is confused and suffers and takes itself to be just a small part of its own form, seriously. It has to play along, meeting people where they are, even when it is clear that it is all just the dance of the awake emptiness. When awakened to itself, this dance is free enough to play along in whatever ways arise.