"всё равно" is more strong.

"Ему всё равно" = "He doesn't care", literally "All these things are equal for him (and equal to nothing)".
So, "я всё равно ухожу" is something like "whatever (did/will) happen (it's all the same - nothing), I'm going".

"всё же" is more like "I have taken your opinion and actions into account, but they still/though don't change my mind".