A delta complex typically requires much fewer simplexes than a simplicial complex to triangulate a space.  For instance, the minimum number of vertices in a delta complex for the torus is 1, whereas the minimum number of vertices in a simplicial complex triangulating the torus is 7. 

A delta complex typically requires much fewer simplexes than a simplicial complex to triangulate a space.  For instance, the minimum number of vertices in a delta complex for the torus is 1, whereas the minimum number of vertices in a simplicial complex triangulating the torus is 7. 

See pgs 102-104 of Hatcher's "Algebraic Topology" for a precise definition of Delta complex. It roughly says that the interior of any simplex is mapped homeomorphically to its image in the space, though interiors of different faces may be identified.

Each simplicial complex is a delta complex whose simplices are uniquely determined by their vertices. This means, in a simplicial complex each n simplex has n+1 distinct vertices, and no other n simplex has this same set of vertices. While in a general delta complex it need not happen, this means many simplices can have same interior and vertex.