I said: "Tomorrow will be our one-to-one meeting with Mr.XYZ."
My friend: "OK, one-on-one."
Which is correct?
One-to-one is used when you talk about transfer or communications. You may use one-to-one when you can identify a source and a destination. For eg., a one-to-one email is one sent from a single person to another, i.e., no ccs or bccs. In maths, a one-to-one mapping maps one element of a set to a unique element in a target set.
One-on-one is the correct adjective in your example. See Free dictionary. One-on-one is used when there are two people involved in mutual exchange, as happens in a meeting.
The difference is rather usage-based, but most important: Do not forget the hyphens.
This might be confusing because of the similar phrases such as "man-to-man" and "heart-to-heart", which refer to an interaction, vs one-to-one, which refers to a transaction (or in the database world, a relationship).
Meanwhile "One-on-One" has a sort of domination/subordination sound to it, where one is on top of the other.
At the end of the day, when two people are having a private conversation or face off or meeting, it's a one-on-one.
Either may be correct, so the tie-breaker is local usage and avoidance of confusion. In my circle, one-to-one connotes a mathematical relationship while one-on-one is used more often in athletic contests. So a meeting might be either. But if you talk about a meeting as one-to-one, the listener may think you are referring to the hour.
One-to-one is less combative, and implies a communication between two people.
One-on-one implies an interaction between two people, and can carry more of a combative subtext, like a one-on-one karate fight.
Face-to-face can also imply there are more than two participants. One-on-one has been used in business meetings for years; and, yes--there is often a "dominance" factor. One is a "leader" or "manager" and one is a subordinate. It's not necessarily pretty, but that's business for you.
Adding point to accepted answer. one-on-one / person-to-person meeting: 1) One person listens to other to know his ideas, suggestions, feedback. One person tries to understand the other persons' expectations, problems and suggestions. One of purposes of this one-on-one meeting would be, each person would like to improve the quality of delivery/contribution to the organization/purpose/business.