For the past couple of years, I have been preaching the Neo4j gospel in many different places, meetups, conferences, and what have you. For the most part, I have been using a very specific demo that has been super well-received: The Belgian Beer Graph. It started out as a learning experience for me personally back in the day when Neo4j did not have any proper data import functionality - and I had to load the graph by jumping through all kinds of hoops.
And now: we can play this demo straight into the Neo4j browser. Such JOY!