You may want to review our licensing page. In particular, for Neo4j Desktop:
Neo4j Desktop is the new mission control center for Developers. It’s free with registration, and it includes a free development license for Enterprise Edition allowing you to use Neo4j Enterprise on your local desktop for developing applications .
Basically it's for local development only, Neo4j Desktop isn't intended or licensed for deployment or usage as a server version (we have the server versions of Neo4j Community and Enterprise for that instead). It can freely be used as a client to connect to a separate server deployment, however.
You may want to review the rest of the licensing page, as there are various scenarios which will allow free Neo4j Enterprise usage (see the FAQ for details).
Thanks! I will try to get the Desktop version working a bit longer before I entirely give up!