Adonijah and Solomon both wanted to be king. What was the difference to the people?
In 1 Kings 1, Adonijah exalted himself. Contrast him to Abraham, David, and even his own son, Absalom. One made himself a king, the other made himself a servant. People are the true earthly power base of any regime. Adonijah tried to impose his rule from above. Solomon went to the people. Big difference.