As Will descibes from another angle, a graphdb will keep your data separated from your application classes and objects. A graphdb also has more built-in functionality to deal with graphs, obviously - like shortest path or deep traversals.
Another important difference is that in a graphdb like neo4j you can traverse the graph based on relationship (edge) types and directions without loading the full nodes (including node properties/attributes). There's also the choice of using neo4j as backend of an object db, still being able to use all the graphy stuff, see: jo4neo This project has a different approach that could also count as an object db on top of neo4j: neo4j.rb. A new option is to use Spring Data Graph, which gives graphdb support through annotations.
The same question was asked in the comments to this blogpost.