First, "date" can be a noun or a verb. If I say "I have a date tonight," it doesn't imply anything about exclusivity or whether I've dated this person before. It could be with someone I've just met or someone I've been dating for a long time. People also talk about having a date with their spouse. "I'm dating someone," implies an ongoing relationship, probably exclusive, and that we have dated before and expect to continue dating. If I say "I'm in a relationship," it definitely implies an ongoing, exclusive commitment.

The words have little in common. You probably could say a boy and a girl who go out on a date（约会） have a relationship（关系）.

A date could be a once off meeting. You may have a date to go out to dinner with someone and if you don't like the person, you would not go out together again. If you have a relationship, then that means you have an ongoing association with someone. You meet for the first time, like each other, go out on a date, then decide to continue seeing each other. If you make a commitment to that person and continue to see them, then you don't go out with other people because you now have a 'relationship'. I hope this is clear and that you now understand the difference.