Hope does not imply belief. But faith is a kind of belief and so, implies belief. Therefore, hope does not imply belief and faith does. So they cannot be the same.
Well said. Hope has to do with a lack of acceptance. If you accepted everything as it is, you wouldn't have hope.
Faith, as it especially pertains to hope, is belief in that which can't be proven, or appears to be untrue... for instance Martin Luther King, Jr's faith that white people could overcome their fears. Such belief has a deep anchor.
Hope is directed toward an ideal. A person with hope is prone to cynicism. The condemnation that's implicit in hope takes over and the ideal is blotted out. This is false acceptance. It's an attempt to ease the pain of being hopeful.
Faith stabilizes hope. An example of that is faith that no matter how bad things seem, it's still right to trust nature.