Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Hanafi means the person who follows the school of Fiqh that is ascribed to Imam Abu Hanifa in the matters of Fiqh and the rules on which that School is based.
Al-Salafi means the one who affiliates himself with good ancestors; they are Sahabah, their followers, and those who were in the three good centuries which are the best of all centuries as certified in Shariah. All four Imams were in this best period. He (al Salafi) also regards the sayings of the good ancestors, follows their understanding of the texts of Shariah and their way and method in faith and rulings of Shariah.
In fact, the four Imams follow al-Salaf the good ancestors in their creed. Those who follow the four Imams actually follow al-Salaf, but there are people who claim that they follow the four Imams but at the same time they have deviated from their method.
We are ordered to believe in Shariah just as the companions of the Prophet (Sallallahu Alaihi wa Sallam) believed without any distortion, corruption, or alteration. This is the method of al-Salaf, hence blessed is the one who adopts this method.
In short, a Hanafi may be a Salafi and may not be the same is true with the other followers of the four schools of thought. It is based on the extent of the agreement and the disagreement of the person with the method of Sahabah and their followers in the matters of faith and creed.
Allah knows best.