There are triplestores (semantic databases), and there are general-purpose graph databases.
Both are based on the similar concepts of linking one "item" to another via a relationship. Triplestores support RDF and are queried by SPARQL, but such add-ons can be (and are) implemented ontop of general-purpose graph databases as well.
What is the fundamental difference that would make you prefer a semantic db / triplestore to a general purpose graph database like neo4j?
Triples stores are graph databases. RDF is a graph. Granted, triple stores tend to not store the information internally as a graph; that's sub-optimal for query answering, but they're still graph databases.
You'd prefer them to something like neo4j if you're interested in an ecosystem based on W3C standards. Makes interop with other RDF-based systems trivial, and it makes picking up everything and moving to a different triple store quite painless.