In Catholic theology, a sacrament is an occasion on which we encounter God in an incarnate way - an outward sign of inward grace. A sacrament both symbolises/signifies an encounter with God, and effects that encounter.
We can, of course, encounter God anywhere, at any time and in any circumstance. But the “seven sacraments” are particular occasions, considered to be instituted by Jesus Christ, on which we are particularly assured of such an encounter.
Also in Catholic theology, marriage is a natural social and human reality. Anyone, Christian or not, can marry. Marriage can be a sacrament, but it does not necessarily have to be. A non-sacramental marriage is, nevertheless, a perfectly valid and entirely real marriage.
One of the conditions on which marriage is seen as a sacrament is the marriage represents the living-out of the spouses’ baptismal vocation. Unless the spouses are baptised, therefore, the marriage is not considered to be sacramental. I think Doreen’s husband’s RCIA instructors were mistaken; a marriage between a baptised person and an unbaptised person is not sacramental. It is not considered to be sacramental for one of them, but not the other. There’s only one marriage here; it’s either a sacramental reality or it’s not.
As to the difference, it’s theological. In practical terms, there isn’t a great difference. A valid natural marriage is entirely valid, and consequently if you’re in a non-sacramental marriage with A, you are not free to marry B in a sacramental marriage. There are some very limited circumstances in which the canon lawyers treat the two kinds of marriage in different ways, but most of us will never encounter them.