What is the difference between a tachi and a katana?

The main differences are curvature, wearing method, and historical era. The tachi has a deeper curve and was suspended edge-down from the waist, suited to mounted combat. The katana has a shallower curve and is worn edge-up through the obi belt for faster drawing. The tachi was the dominant sword from the Heian through early Muromachi periods, while the katana became standard from the mid-Muromachi period onward.