According to my dictionary have an appointment and have an engagment mean the same thing but I figure there must a difference? Is have an appointment more used in business? Thanks in advance. Frank
You can make/arrange an appointment to see someone in a business context. You can make an appointment to see/visit the doctor/the dentist/the optician.
Engagement is a little less precise because when someone says: I have an engagement/a prior engagement on that day/at that time, it could be a social meeting or a business
meeting. To summarise, you make an appointment in order to arrange a time and place to meet someone but when you have an engagement, it simply means that you are not free or that you are not available at that time and you are not saying what sort of activity/meeting it is.