Recently I have been looking into graph databases like Neo4j and into logic programming in Prolog and miniKanren. From what I have learned so far, both allow specifying facts and relations between them, and also querying the resulting system for some selections. So, actually I cannot see much difference between them in that they both can be used to build a graph and query it, but using different syntax. However, they are presented as totally different kinds of software.
Except the technicality that databases maybe propose a more space-time effective storage technology, and except that tiny logic cores like miniKanren are simpler and embeddable, what is the actual difference between graph databases and logic programming languages, if they are both just a graph database + query API?
No, logic programming as embodied by those things and neo4j are quite different.
On one level, you're right that they conceptually both amount to graph storage and graph query. But for logic programming, it's only conceptually graph query, there's no guarantee that it's actually stored that way (where with neo4j, it is).
Second, with logic programming you're usually trying to establish horn clauses that allow you to reason through lots of data. You can think of a horn clause as a simple rule, like "If a person is male, and is the direct ancestor of a biological child, that implies that person is a father". In cypher with neo4j, you would describe a graph pattern you wish to match, that results in data, e.g.:
MATCH (p:Person)-[:father*]->(maleAncestor:Person)
RETURN maleAncestor
This tells to traverse the graph by father relationships, and return male ancestors. In a logic programming language, you wouldn't do it this way. You might specify that a being a father of b means that a is male, and a is an ancestor of b. This would implicitly and transitively state that for all valid a/b pairings. Then you'd ask a question, "who are the male ancestors"? The programming environment would then answer that by exploiting your rules. That would have the effect of building a traversal through the data that's very similar to the cypher I specified above, but the way you go about understanding your data and building that traversal is totally different.
Logic programming languages usually work via predicate resolution. A graph query language like cypher works by a combination of pattern matching, and explicit path designation. They're very different.