Even though these two words/concepts are virtually interchangeable, there is a difference between them which I will discuss in this article beginning with some definitions from The Australian Pocket Oxford Dictionary:

Faith: trust; belief in religious doctrine or divine truth; religion; loyalty, fidelity, confidence.
Trust: firm belief that a person or thing may be relied upon, state of being relied upon.

The dictionary definition of Faith includes Trust, so yes, they are synonyms. However, Faith is more often than not associated with the belief in something spiritual, or divine that cannot be ‘proven.’ This sustains people through very challenging life experiences (i.e. God has a plan; God will provide; there is a Divine Purpose for everything, etc.).
The following prayer illustrates this, the last sentence, in particular:

“To get something you never had, you have to do something you never did. When God takes something from your grasp, s/he is not punishing you but merely opening your hands to receive something better. The Will of God will never take you where the Grace of God will not protect you.”

Trust is more about the belief you have in an individual, including yourself – the “state of being relied upon” is a measure of ‘proof’ demonstrated by her/his/your actions. That is, when s/he lets you down, you may have a difficult time trusting that being again, or, trusting yourself for making an error in judgment especially when this occurs time and again. Trust may be demonstrated in interpersonal relationships in the following ways: by keeping a secret, or confidence, supporting someone during both good and bad times, and, by being truthful.

Of course, you can still say that you have Faith in someone (as evidenced by the words loyalty, fidelity and confidence in the dictionary definition), as well as saying that you Trust God, the Universe etc., to provide for and protect you. Either way, without this concept in both human and spiritual relationships, life would be devoid of Meaning.

Some ways to Keep the Faith: Read inspirational material; visit your place of worship which may be a religious/spiritual building, or, a spot in Nature; talk to a wise, spiritually aware individual. A way to measure self-trust involves paying attention to your body when you are thinking of an individual, or situation that you are unsure about: an uplifting, warm bodily sensation, where you take in a breath spontaneously, is a ‘Yes’; a sinking, low, heavy physical feeling, that may cause a sickening feeling in your gut, is definitely a ‘No.’

Aromatherapy oils that promote Faith: Rose, Frankincense and Cedarwood.
Bach Flower Essences that promote Faith: Gorse and Gentian.

Happy Healing!