Your question implies there is a conflict between these two men in The Crucible; however, I'm not sure I would agree with the premise that they actually have one. They do, however, have points of disagreement. Certainly they are not in agreement that going to church and having children baptized by a man who is not respected is still the right thing to do. Certainly they view the testimony of the girls in different ways--Proctor knowing they're lying to protect themselves, Hale believing them because he is deceived by the false signs of witchcraft. Certainly they both want to respect the Court and the law, until it turns on Proctor and he condemns the entire system--as does Hale, eventually. Certainly Hale begs Proctor to tell a lie in order to save his life, which of course Proctor cannot, in the end, do.
What they have in common is certainly just as powerful as where they diverge. They both believe a covenanted man in this society should know certain catechisms of the church, such as the Ten commandments. They both have a strong faith in God, though each of them wears it differently. They both want the truth to be told in Salem before man and God. In the end, this is what makes them more similar than different.
In Act one, Proctor expresses a sentiment to Reverend Hale soon after the cleric's arrival in Salem, which would set the foundation of their conflict later in the play:
I've heard you to be a sensible man, Mr. Hale. I hope you'll leave some of it in Salem.
When the reverend later shows up at their residence and declares that Rebecca Nurse is being suspected of witchcraft, Proctor states that, in terms of what he had said earlier, that the reverend cannot surely believe that she is guilty, (being a 'sensible man'). Hale's response confirms that he has been convinced by the preponderance of evidence that the Devil, was indeed, launching an attack on the village. He asks whether Proctor accedes to this fact.
Proctor expresses doubt about Rebecca's guilt whilst the reverend insists that the Devil is undeniably sly and states that Rebecca has not been accused yet. He states that the purpose of his visit is to question Proctor and Elizabeth's Christian character. This request would form the focal point in their conflict. John resents the implication and states that they are not afraid of being questioned.
It is then that the reverend starts questioning the Proctor's commitment to the church and their belief. His assertions are all based on evidence provided by the reverend Parris. Proctor is clearly uncomfortable and indignant about Parris' accusations and vehemently defends himself each time reverend Hale questions his behaviour, such as the number of times he has been attending church as well as the fact that only one of his children have been baptized.
John's intense disgust with what Parris represents comes to the fore. He states that he owes the reverend no favors and is completely against his materialism, stating that Parris is more interested in acquiring earthly goods than providing for his community's spiritual needs.
It becomes apparent, however, that reverend Hale has greater faith in reverend Parris and states that any misdemeanor against the church, or disobedience, creates a chink in the armor of religion against the forces of evil. Proctor eventually comes straight to the point and wants to know exactly what reverend Hale is suggesting about the Christian nature of his household. He assures the reverend that they do not bear any love for Satan. Reverend Hale does not seem quite sure about Proctor's claim and rather states that he 'prays dearly' that it is so.
The conflict between the two is accentuated later when Elizabeth urges John to tell the reverend what he knows. When John speaks about Abigail having told him that the children's sickness had nothing to do with witchcraft, the reverend is startled and wants to know why John did not declare this earlier. John states that he did not know then that things would get so out of control. Reverend Hale argues that he had examined others and they confessed to dealing with the Devil. He implies that John is lying and asks if he will testify to this in court. He believes John is faltering but Proctor assures him that he will testify if necessary.
Reverend Hale is even more surprised when Elizabeth states that she does not believe in witches if she is accused of being one. This shakes the reverend because her statement goes against the Gospel. He is confronted again by Proctor who declares that his wife does not doubt the Gospel and that they run a Christian house.
The conflict worsens when Cheever later comes to arrest Elizabeth on a charge of witchcraft based on testimony provided by Abigail Williams. Proctor tears up the warrant of arrest and refuses that Elizabeth be taken. She however, intervenes and says that she must go. Proctor is overwhelmed since reverend Hale has stood by and allowed the ignominy to happen even though, as a 'sensible man,' he should have seen that the evidence about the poppet was malicious and fabricated after Mary Warren's testimony. John calls him a 'broken minister and 'Pontius Pilate'.
After Elizabeth's removal, John demands that the reverend get out of his sight. Hale states that he will testify about what he has witnessed and beseeches John to show charity. He states that John cannot blame the vengeance of a little girl (Abigail) as the reason for what has happened. John, however, is overwhelmed and angry and shouts at him:
You are a coward! Though you be ordained in God's own tears, you are a coward now!
Hale then leaves.