What is the difference between a Graph Database and a Network Database?
Copying from the book Designing Data-Intensive Applications by Martin Kleppmann.
In the network model, the database had a schema that specified which record type could be nested within which other record type. In a graph database, there is no such restriction: any vertex can have an edge to any other vertex. This gives much greater flexibility for applications to adapt to changing requirements.
In the network model, the only way to reach a particular record was to traverse one of the access paths to it. In a graph database, you can refer directly to any vertex by its unique ID, or you can use an index to find vertices with a particular value.
In the network model, the children of a record were an ordered set, so the database had to maintain that ordering (which had consequences for the storage layout) and applications that inserted new records into the database had to worry about the positions of the new records in these sets. In a graph database, vertices and edges are not ordered (you can only sort the results when making a query).
In the network model, all queries were imperative, difficult to write and easily broken by changes in the schema. In a graph database, you can write your traversal in imperative code if you want to, but most graph databases also support high-level, declarative query languages such as Cypher or SPARQL.
The network databases like CODSASYL are still more or less based on a hierarchical data model, thinking in terms of parent-child (or owner-member in CODASYL terminology) relationships. This also means that in network database you can't relate arbitrary records to each other, which makes it hard to work with graph-oriented datasets. For example, you may use a graph database to analyze what relationships exist between entities.
Also, network databases use fixed records with a predefined set of fields, while graph databases use the more flexible Property Graph Model, allowing for arbitrary key/value pairs on both nodes/vertices and relationships/edges.