Inspired by a Simpsons quote (paraphrased),
Wiggum: You have that much faith in me?
Homer: No, faith is what you have in things that don't exist. Your awesomeness is real.
I will be writing as if I am referring to people, though you can have faith in or trust something as much as someone.
In the most basic terms: Faith is believing that someone will aid you. No reason to really, no guarantee, no need. Just you believe this. Pretty much the exact opposite of the scientific method*. It doesn't have to be absolutely certainty--though that is what absolute faith is. Even without evidence, the bedrock-solid knowledge that this fact is true.
Flip side, used less: Knowing that it will all go wrong fits equally well. Knowing that the person with the wrong scar over the right eye is going to stab you in the back. You don't know the person. You just see the face.
Laconic: This I believe.
Trust is different. You might have faith in some people after not seeing them for years, but trust has to be kept up. Help you up; stand by you when no one else would; just be there when you need them to be and you trust them. You have seen how they acted in the past, and it leads you to believe they are good people. You know they'll help you and yours.
Flip side, used less: Pretty obvious at this point. Push you down; run when someone else stood with you; leave when you need them most. Worse, do that and then expect you to help them. Maybe even worse than that. Because you will, and tomorrow they're back to this knife-twisting trust, you know they'll hurt you. But you choose. Do you turn you back on you own ideals? Do you stick to them, cruel people or no?
Or do you change your ideals for a new situation?
*I am not belittling the scientific method or faith. The scientific method does require setting one's faiths aside for the sake of seeing reality as it is--but not everything is scientifically provable. And stuff that isn't scientifically proven may well still be true.
Dictionary, closest to mine posted:
Faith: b (1) : firm belief in something for which there is no proof (2) : complete trust
Trust: a : assured reliance on the character, ability, strength, or truth of someone or something