In my opinion - you don't only miss out more exploring of the town, but all of the unique island. Naxos is the largest of the Cyclades islands, and has so much to offer. Landscape, villages, archeological sites, some of the best beaches in Greece and much more. In addition to this - given the time - you also have the opportunity to visit neighbouring islands.
To give you an idea - look up this site and use the drop down menues:
You may also get en idea by looking up my albums:
I agree with Greek-Traveler. It would take many days just to explore the island which is filled with wonderful sites, vistas, beaches, villages, mountains, centuries-old Byzantine Churches, great food and so much more.
We were there for 8 days and still could have stayed longer. We did, however, do a day cruise to Delos/Mykonos but we had the extra time . . . with 3 days you don't.
With your 3 nights I would recommend staying on land and explore the island. It's a great combination of Old & New World Greece.Edited: 09 August 2017, 15:32
We had 4 days and spent 1 day doing a day cruise around the island, it was pleasant, but nothing special. There wasn't much to see. We did one on Milos and it was exceptional because the landscape was spectacular.