Planes leave from Concourses, not Terminals (and International departures can technically leave from any Concourse). The Terminals are where passengers arrive and depart, check in, go through security, etc. Once you are past security at either Terminal in ATL (or immigration if arriving Internationally), you are free to move about the system to any Concourse, because they are all interconnected via the PlaneTrain. .

The only exception to this is passengers who arrive internationally and whose final destination is Atlanta. Those passengers are not allowed to board the PlaneTrain, and must exit the airport via the International Terminal. The other exceptional condition is that passengers arriving for International flights other than Delta must check their bags at the International Terminal (that obviously does not apply in the instant case, since the OP is flying Delta and is not checking bags).

So if you are flying out on Delta, you can arrive at either Terminal. If you are flying out on any other airline, you must arrive at the appropriate Terminal for your destination (Domestic or International). If you are arriving Domestically, you leave the airport via the Domestic Terminal, and if arriving Internationally (and your destination is Atlanta), you leave via the International Terminal.

I guess someone who was arriving Internationally and had a boarding pass for onward travel within the US could leave via the Domestic Terminal to visit downtown on a long layover, but I've obviously never had occasion to try that.

-JimG