The law says the punishment for adultery is death. Moses gave reasons for divorce and remarriage. Jesus said "It was because of the hardiness of your hearts" Jesus called divorce and remarriage adultery, but never called for any punishment. Are there different types of adultery. It seems the two are treated differently.
Moses probably did not intend a man may divorce his wife for any reason, however his instructions were abused such that men were easily able to divorce their wives over silly reasons. It is this Jesus was likely referring to when He said "hardness of your hearts".
"If a man marries a woman who becomes displeasing to him because he finds something indecent about her, and he writes her a certificate of divorce, gives it to her and sends her from his house, and if after she leaves his house she becomes the wife of another man, and her second husband dislikes her and writes her a certificate of divorce, gives it to her and sends her from his house, or if he dies, then her first husband, who divorced her, is not allowed to marry her again after she has been defiled. That would be detestable in the eyes of the Lord. Do not bring sin upon the land the Lord your God is giving you as an inheritance" (Deuteronomy 24:1-4 ).
In the Old Testament, ancestry and genealogy (the Jewish bloodline) mattered because the Jews were God's chosen/favored people. Adultery could mess up bloodlines etc., not only through outsiders but also amongst the Jews as there were different clans. In addition to the sin of adultery itself, messing up bloodlines probably called for the death penalty. However, ancestry and genealogy do not matter much (or at all) in the New Testament, which kind of follows no physical punishment for adultery including remarriage adultery. Adultery is still serious nonetheless, even allowing for divorce, and considered a sin. However, no physical punishment is required for the physical act. God cares more about our mind/heart in the New Testament (thus, adultery of the heart). If we applied the physical punishment, most everyone would be punished.