This post addresses only the Holy Bible, but the title is true, regardless of religion.
Simply put, belief is limited by function. In particular, no function is necessary with belief. We can believe whatever we want, mainly due to our literal free agency. I can believe there are Na'vi [tall, slender, blue-skinned humanoids] on the moon, Pandora, but I am not compelled to do a bloody thing about that belief, or any other. Belief can be a simple noun or verb, "believe."
However, the distinction of faith, as opposed to belief, is that faith demands our personal commitment to action, or it is not faith. Above, belief was shown to be, as for personal commitment, more noun than verb.
Both words are used in scripture. However, they cannot and should not be interchanged by random choice. In fact, in Greek, in the New Testament, at least, [Koine Greek is the apparent original written language], words are used to try to equate the use of one to mean the other. Examples:
- Philippians 1:29 "For to you it has been granted on behalf of Christ, not only to believe in Him, but also to suffer for His sake," In Greek, the bolded Greek word is: πιστεύειν [pisteuein], "to have faith." This translation is substantiated by the added phrase "... but also to suffer for his sake." adding action to the conviction by faith.
- I John 4: 1 "Beloved, believe not every spirit, but try the spirits whether they are of God: because many false prophets are gone out into the world." In Greek, the bolded Greek word is: πιστεύετε [pisteuete], "do not believe." This translation 'believe" ignores the Greek root pisteu... which should have translated as "Have not faith," that is, do not act on the word of every spirit...
- 2 Thessalonians 2:13 "...because God chose you⁺ from the beginning unto salvation in the sanctification of the Spirit and belief of the truth" In Greek, the bolded Greek word is: πίστει [pistei], "[by] faith This translation 'believe" ignores the Greek root pistei... which should have been translated as "[in] faith."
All of these translations of the Bible where "faith" ought to be the word used, use instead "belief," or proper noun/verb distinctions of same. This is not correct, because the Greek root πιστεύ [pisteu] means expected action on faith in every instance. It is by personal action on the suggestion given that confirms the truth of that in which faith was applied.

