Belief is the conviction of the truth of a statement or the reality of a being or phenomenon. Faith, according to the dictionary, is both belief and trust in something. Thus, one can believe in something, but not have faith in it.
I most often hear the terms faith and belief in conversations about the supernatural. People believe in ghosts, aliens, fairies, deities, and whatnot. But do people have faith in these things? Some do. Some people have faith in God--they believe not only in God's existence, but in His/Her/It's omnipotence and omnipresence. They trust that God is in charge.
A problem develops when one has belief, but not faith, in God. Is it possible? Sure. Is it logically defensible? Possibly, based on one's concept of God. God does not have to be the puppet master, pulling the strings of humanity. God, or the Universal Essence, or the Thing Greater than Us, could just be nudging us, giving us clues as to the right direction. Or maybe God is a passive observer, seeing all, perhaps judging all, but not interfering.
I do believe in a Force Greater than Myself. Can I call it God? Not really, because my belief does not fit neatly into the boxes that God has been placed in. I do not believe in a Christian, Muslim, or Jewish version of God. I would call the Force I believe in a sort of universal subconscious, a collection of the spirits of all things living, dead, and unborn, all their experiences and energies connected in a network that spans time and space. I imagine it as a sort of immense web, where vibrations from each part can be felt anywhere. We are each a part of it, and conversely, it is a part of each of us. This is my spritual belief--no one gave it to me, or helped me to come up with it. It is simply what I feel in my heart to be true.
But do I have faith in it? No. I don't believe that this universal subconscious is there to protect people or watch over us. I don't have to trust in it. There isn't anything to trust in. It just is. Ultimately, I am still in charge. I can't blame or thank any higher power for my life as it is. I regard this as the truth, and no amount of evangelism or preaching has ever been capable of changing my mind. And because of this, life is occasionally overwhelming.
It would be easier if I could truly believe that "God does not give us more than we can handle." But that's not how my belief works. God is not doling out experiences, tragedies, and triumphs. They occur based on my own actions, the actions of others, with a touch of entropy thrown in to the mix. It is an uncomfortable belief. But it is mine.