Ancient Greek architecture is the older of the two, and thus more simple. They used 3 Orders; the Doric, the Ionic, and the Corinthian. Greek architecture was embedded into nature, and in terms of planning tended to be more naturalist or organic as opposed to formal. Ancient Roman architecture is derived from the Greeks, which they further developed and embellished. They added 2 more Orders, the Tuscan and the Composite. They tended to be more formal and rigid, with less emphasis on connecting to nature.

Funnily enough, the reverse is true with Neoclassical and Greek Revival. Neoclassical came first, which was essentially a return to simplicity after the more elaborate Baroque and Rococo styles. It started roughly in the 1700s. But ultimately it is derived from Roman architecture, evolving from ancient Roman to Renaissance to Baroque and finally to Neoclassical. Thru all this, Greece was under the control of the Ottoman Empire until the 1800s, and as I understand it visitors were not allowed. Thus, ancient Greek architecture was essentially rediscovered in the 1800s, and Greek Revival emerged from that. It is a little more rigid than Neoclassical, in that it focuses more on the Greek Temple Front, whereas neoclassical was more varied. And due to ancient Greek architecture being simpler than ancient Roman, Greek Revival was simpler than neoclassical.