Yes, the API seems like the major difference, but is not really a superficial one. Conceptually a set of objects will form a graph and you could think of an API that treats this graph in a uniform way. Conversely, you could in theory mine a generic graph structure for patterns and map them to objects exposed via some API. But the design of the API of an actual product will generally have consequence on how data is actually stored, how it can be queried, so it would be far from trivial to, say, create a wrapper and make it look like something else. Also, an object-oriented database must offer some integrity guarantees and a typing structure that a graph database won t normally do. In fact, serious OO database are far from "free form" :)
Take a look at [HyperGraphDB][1] - it is both a full object-oriented database (like db4o) and a very advanced graph database both in terms of representational and querying capabilities. It is capable of storing generalized hypergraphs (where edges can point to more than one node and also to other edges as well), it has a fully extensible type system embedded as a graph etc.
Unlike other graph databases, in HyperGraphDB every object becomes a node or an edge in the graph, with none-to-minimal API intrusion and you have the choice of representing your objects as a graph or treating them in a way that is orthogonal to the graph structure (as "payload" values of your nodes or edges). You can do sophisticated traversals, customized indexing and querying.
An explanation why HyperGraphDB is in fact an ODMS, see the blog post Is HyperGraphDB an OO Database? at Kobrix s website.