Difference between Engaged to someone and Engaged with someone

engaged to someone—bound by a promise of marriage to a person:

• Mary was engaged to Jim for three months before their wedding.

engaged with someone—

1. having one’s time occupied by a person:

• You cannot see Mr. Brown just now, as he is engaged with a visitor.

2. involved in hostilities with enemy troops:

• They could scarcely have reached the camp before they found themselves engaged with the enemy.