Trust' indicates a depth and a sense of assurance that is based on strong but not logically-conclusive evidence, or based on the character, ability, or truth shown by someone or something over time or across situations.
Faith implies that there’s not necessarily a proven record, and in with religious faith there’s none at all. This means that you’re taking it “on faithâ€. You have no reason to believe that anything a religion says is true, it’s all faith.