The difference between "only one" and "one and only one"

In ordinary conversation, only one has the same meaning as one and only one. The shorter phrase is used almost every situation.

In mathematical logic, it's often desirable to make a distinction between zero or one and exactly one. In that situation one and only one is used to indicate that the count cannot be less than one or more than one.

The meaning of both "only one" and "one and only one" is the same. However, "one and only one" adds emphasis to the fact that there is only one, and draws attention to it. For example, the student who is the only one who failed, might feel more ashamed if the teacher uses "one and only one", as the teacher might be perceived as purposely drawing attention to that fact, for whatever reason.