Difference between a graph and a hypergraph database?

How a certain graph database handles its edges is an implementation detail. Hence an answer cannot really be given in regards to "[hyper]graph databases in general".

From the point of mathematical graph theory however there is a difference:

• Edges as known from standard graphs model (directed or undirected) 1:1 connections.
• Hyperedges as known from hypergraphs model (directed or undirected) n:n connections.

Graph vs. Hypergraph:

A simple graph can be considered a special case of the hypergraph, namely the 2-uniform hypergraph. However, when stated without any qualification, an edge is always assumed to consist of at most 2 vertices, and a graph is never confused with a hypergraph. ^(Source)

Undirected hyperedges:

A[n] [undirected] hyperedge is an edge that is allowed to take on any number of vertices, possibly more than 2. A graph that allows any hyperedge is called a hypergraph. ^(Source)

Directed hyperedges:

Directed hypergraphs (Ausiello et al., 1985; Gallo et al., 1993) are a generalization of directed graphs (digraphs) and they can model binary relations among subsets of a given set. ^(Source)