The difference between a heart of stone and a heart of flesh us the same as the difference between spiritual death and spiritual life. The terms are contrasted in Ezekiel 11:19 and 36:26 where the heart of stone is used to mean a person is spiritually dead.
“And I will give them one heart, and put a new spirit within them. And I will take the heart of stone out of their flesh and give them a heart of flesh.”Ezekiel 11:19
“Moreover, I will give you a new heart and put a new spirit within you; and I will remove the heart of flesh.”Ezekiel 36:26
In both Ezekiel 11:19 and 36:26, God says he will put a new spirit within the people. In light of this, he says he will remove the heart of stone and give them a heart of flesh. Therefore, the heart of stone must signify being spiritually dead, and the heart of flesh must mean being spiritually alive.
We see this supported in the book of 1 Samuel that speaks of the heart dying and becoming like stone.
“But in the morning, when the wine had gone out of Nabal, his wife to,d him these things, and his heart died within him so that be became as a stone,1 Samuel 25:37
Put my spirit within you
The term “put my spirit within you” occurs in Ezekiel 36 27; 37:14
“I will put my spirit within you and cause you walk I my statutes, you will be careful to observe My ordinances.”Ezekiel 36:37
I will put my spirit within you and you come to life, and I will place you on your land. Then you will know that I, the Lord, have spoken and done it,, declares the Lord,” Ezekiel 37:14.
In these two verses, when God pits his spirit within someone they will walk I. His statutes (36: 27) and they will come to life (37:14).
Heart Like Stone
The idea of the heart becoming like stone clearly signifies a deadness of heart and the unwillingness to follow God.
In Zechariah 7:12 it says they made their hearts like flint so that they could not hear the law and the words which the Lord of hosts has sent by His spirit through the former prophets; therefore great wrath came from the Lord of hosts.
As stated at the beginning of this post, the contrast between the heart of stone and the heart of flesh signifies being spiritually dead and spiritually alive. In addition to having a heart of flesh means that God has put his spirit within a person. So it is speaking of the work of God awakening a person spiritually so that they might understand him and follow him.