Felicitate, congratulate mean to express one’s pleasure in the joy, success, elevation, or prospects of another.
Felicitate is the more formal term and carries perhaps a stronger implication that the person who felicitates regards the other as very happy or wishes him happiness.
Congratulate is the more common and often more intimate term; it usually implies that the congratulator regards the other as a person to whom good fortune has come or on whom fortune smiles.