What are the differences or beliefs which distinguish these three types of priests?
There is no difference regarding what they believe. They are Catholic, after all.
Each order has a particular charism which benefits the Church. Franciscans were founded to imitate the poor Christ and to minister to Him in the poor. Jesuits were founded to be the Church’s militia, ready to serve the Church however needed. Dominicans were founded to imitate the preaching Christ, preaching the Gospel with theologically sound doctrine to all who will hear.
Certainly, all those who live religious life make a vow of poverty. All are committed to serve the Church. All priests preach. But each of the orders mentioned above specialize in their charisms.
Fr. Vincent Serpa, O.P.