"Claiming ACID" what is ambiguous about that? Transaction support with different serialization levels, like other databases that offer it.
And Neo4j originally started b/c RDBMS was not able to execute the complex deep traversals needed in real time. Dedicated storage & query engine for graphs allow you to run statements quickly that would otherwise take too long to execute.
Regarding the data model, the property-graph model is much closer to the object model but with richer relationships, it doesn't suffer from the object-rdbms impedance mismatch and is better suited to express real-world domains & scenarios. It also represents semantic relevant relationships as first class citizens in the database, allowing for proper information representation and much faster retrieval.
Disclaimer: I work with/for Neo4j, for 8+ years and still love it.