You use awoken in the various ‘perfect’ tenses; that is, the present perfect, the past perfect, and the future perfect.

You’re probably going, “WHAT?” In an example, you say: I had awoken from a bad dream yesterday.

Awaken is a more passive verb, which means you can use it but your sentence will be less interesting. If you still wish to go there, then you use awaken in such sentences as: I was awakened by the thunder, I had been awakened by the alarm.