In S03E23-24 of The Mentalist, Patrick Jane killed a man named Timothy Carter, thinking he was the real Red john, in a shopping mall with a gun hidden in his pocket. Then he waited in the mall peacefully and surrendered to the police. He was not afraid of getting caught.
In S06E08, when he faced the real Red John, he had a gun in his hand yet he did not use it to kill him. One gunshot would have made sure that Red John was killed, once and for all.
So, is there any reason behind him not using his gun? Why take the risk?