I used a graph database in a previous job. We weren’t using neo4j, it was an in-house thing built on top of Berkeley DB, but it was similar. It was used in production (it still is).
The reason we used a graph database was that the data being stored by the system and the operations the system was doing with the data were exactly the weak spot of relational databases and were exactly the strong spot of graph databases. The system needed to store collections of objects that lack a fixed schema and are linked together by relationships. To reason about the data, the system needed to do a lot of operations that would be a couple of traversals in a graph database, but that would be quite complex queries in SQL.
The main advantages of the graph model were rapid development time and flexibility. We could quickly add new functionality without impacting existing deployments. If a potential customer wanted to import some of their own data and graft it on top of our model, it could usually be done on site by the sales rep. Flexibility also helped when we were designing a new feature, saving us from trying to squeeze new data into a rigid data model.
Having a weird database let us build a lot of our other weird technologies, giving us lots of secret-sauce to distinguish our product from those of our competitors.
The main disadvantage was that we weren’t using the standard relational database technology, which can be a problem when your customers are enterprisey. Our customers would ask why we couldn’t just host our data on their giant Oracle clusters (our customers usually had large datacenters). One of the team actually rewrote the database layer to use Oracle (or PostgreSQL, or MySQL), but it was slightly slower than the original. At least one large enterprise even had an Oracle-only policy, but luckily Oracle bought Berkeley DB. We also had to write a lot of extra tools – we couldn’t just use Crystal Reports for example.
The other disadvantage of our graph database was that we built it ourselves, which meant when we hit a problem (usually with scalability) we had to solve it ourselves. If we’d used a relational database, the vendor would have already solved the problem ten years ago.
If you’re building a product for enterprisey customers and your data fits into the relational model, use a relational database if you can. If your application doesn’t fit the relational model but it does fit the graph model, use a graph database. If it only fits something else, use that.
If your application doesn’t need to fit into the current blub architecture, use a graph database, or CouchDB, or BigTable, or whatever fits your app and you think is cool. It might give you an advantage, and its fun to try new things.
Whatever you chose, try not to build the database engine yourself unless you really like building database engines.