I had always thought belief and faith were the same thing. The following example really helps you understand however that there is a difference.
When you approach a chair, you look at it. You analyse its structure, and you brain no doubt completes many calculations whilst figuring out if that chair will support your weight. Once you have collected enough evidence that the chair looks like it will support your weight, you come to the belief that yes it will support you.
Now comes the part that highlights the difference betweek belief and faith. You have looked at a chair, and based on some factors have now come to the belief that the chair will support your weight. You BELIEVE it will support you. Now comes the part where you SIT on the chair! THAT is FAITH in action.
Your brain can analyse the evidence but can only take you to a certain point. Looking at Jesus, you can look at the historical evidence, as well as the biblical evidence to come to a belief in his existance, however belief if not enough on its own!
Even the devil believes in Jesus (James 2:19). Intellectually the devil believes in Jesus, in God, in the Bible! But the devil is not saved! He has no FAITH in Jesus.
When you become a Christian, you first believed, it was then you excersised faith. Faith is the hand that receives the gift of salvation. Faith is a gift from God (Ephesisans 2:8). By grace we have have been saved.