The two words are not synonymous if we are discussing the salvation of men. Faith at least in part is defined as believing that God is (Heb. 11:6). Jesus taught that if a person did not believe that He was the Son of God that person would die in their sins (John 8:24). Part then of faith is believing. The confusion created in the religious world comes about because individuals tamper with the meaning of belief. For some it means just to accept the existence of God and His Son. If one accepts the truth that Jesus died for them then the religious would say, that is all that is necessary. Those who practice “fake healing” when confronted with their failure fall back on that the person who was being healed did not have enough faith. To them then the problem is using the word believe to demonstrate faith. If you do not get healed it was because you did not believe strong enough.
Jesus taught that men must confess him in order to possess a hope for heaven (Matt. 10:32). There were some Pharisees who believed in Jesus. Their belief was not strong enough to cause them to confess Jesus before others (John 12:42). If they did not confess Jesus before men surely the Word of God shows they were not right with God. The second part of Hebrews 11:6 sheds light to remove the darkness created by man. It states there that God rewards those who diligently seek Him. Paul argued with some philosophers that God is near and that men should seek Him and be able to find Him (Acts 17:27).
God requires a diligent search of Him and His ways. A belief that does not push one into action is not a belief that can save. Agrippa believed but he refused to be immersed. He was almost convinced but he was altogether lost. Coming to a knowledge of God can lead one to believe. A constant search of the knowledge can lead one to faith. You cannot have belief without faith and you cannot have faith without works to prove that faith.