What's the difference between an argument and a discussion?

Is an argument only an argument if it contains logic?

An argument, in the formal/academic sense, requires reasons and generally an attempt to persuade. It does not necessarily require logic. There are plenty of terrible arguments out there.

Or rather, at least one premise and its conclusion?

This is getting into what is known as a deductive argument. I would defer to the philosophy books on the topic that state what constitutes such an argument.

Can a group have a meaningful discussion without there being logic and reasoning involved?

I think this is a matter of opinion. A couple expressing their love for one another by discussing feelings can be very meaningful to the couple. No reasons needed. However, I would be less charitable to discussions where emotion should play little to no role.