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What is the difference between
ONE: 
ONE AND ONLY ONE: 
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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.
Chen notation uses a diamond and connecting lines with symbols to describe relationship and cardinality, while Crow's Foot just uses lines with symbols on the end.
A multiplicity of one and a mandatory relationship is represented by a straight line perpendicular to the relationship line. A multiplicity of many is represented by the three-pronged 'crow-foot' symbol. An optional relationship is represented by an empty circle.
A crow's foot graphics represents “many” or “infinite”. A dash (closest to the entity in ERD) represents “one”. An optional relationship is represented by an empty circle. A mandatory relationship is represented by a line.
I think n.piskunov is on the right track.
I would use it as the mathematical analogy found here: https://math.stackexchange.com/questions/247380/analogy-between-one-and-only-one-and-if-and-only-if
"existence of at most one such X that satisfies Y."
As an example, consider these relations: 
A student, Alice, can only have one dorm room at at time. A dorm room can only house one student at a time (for the sake of this example). Next year, Alice will be assigned a new dorm room, and at that point her dorm room from this year will be assigned to a new student.
Alice can have one and only one login (e.g. a11235) and that login can only be assigned to Alice. When Alice graduates, no one else can be assigned the login a11235.
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