The nature at the bare bone level so far from my understanding is.....
Similarities....
1) Act as the Will of the Planet to Prevent the Original Calamity from Returning.
2) Preserve life on the Planet.
Differences....
1) Zodiark preserve life on the Planet by sacrificing the newer generation to preserve those of the past while Hydaelyn preserve the newer generation but sacrifice the past to give future generations a chance to prosper.
Both are designed with one goal being the preservation of the planet but who they priorities differs which is the source of there eternal conflict with Zodiark focusing on the Past and Hydaleyn focusing on the Future.
I can only assume the only time Zodiark and Hydaelyn will ever get along is if they have to defeat a common enemy that threatens the entire Planet which looks like will be born very soon...
Look at.....
Zeno and his plan to consume both Zodiark and Hydaelyn so he can become a being so powerful it would create the world most devastating and bloodiest battle in the history of the Planet against the Warrior of Light just for his person thrills that may end up causing the Greatest and most destructive Calamity not even the Ascians would dare create due to ebing of equal or greater destruction of the Original Calamity that almost destroyed the Planet.