"At this, all Adonijah's guests rose in alarm and dispersed. But Adonijah, in fear of Solomon, went and took hold of the horns of the altar." NIV translation
When they heard of Solomon's coronation, the people scattered but Adonijah went to the tabernacle and grabbed onto the horns of the altar. From what we know, he was not a religious man but grabbing hold of the altar was a plea for mercy.
"Then Solomon was told, 'Adonijah is afraid of King Solomon and is clinging to the horns of the altar. He says, "Let King Solomon swear to me today that he will not put his servant to death with the sword."' Solomon replied, 'If he shows himself to be a worthy man, not a hair of his head will fall to the ground; but if evil is found in him, he will die.' Then King Solomon sent men, and they brought him down from the altar. And Adonijah came and bowed down to King Solomon, and Solomon said, 'Go to your home.'" NIV translation
Unlike Adonijah, King Solomon was a man of God and so he showed mercy upon Adonijah.He came and bowed down to Solomon and it was finished as it was apparent to all that there was only one king and it was Solomon.