Sayounara is something u say to something if ur saying goodbye but your not going to see them again for a while or a long time.
Ja Ne is a more casual goodbye, and so its used more frequently. Its what you'd say if you'd see them again, or even if ur unsure, its more casual but not a goodbye thats too friendly even if u don't know them well, so it wouldn't be rude. But also, Ja Ne can be said differently. It can be said "Jaa, Mata Ne" or just "Mata Ne" or just "Ja Ne" and they'd all basically mean the same thing, just some people say it different ways in Japan, like how we say goodbyes many different ways.
An even more casual goodbye is Bai Bai (aka Bye Bye), they say it the exact same way we do and to them its more like "Cya" or "later".
So also to answer ur question, Ja Ne would be more formal than Bai Bai. Sayounara is not more formal than Ja Ne, but its only used if ur parting with someone for quite a while.