Many thanks to Allen in turn for his generous citation to me. What I had to say is fairly standard, yet I don't think that I'd ever be able to write a book like Hatcher's Algebraic Topology. Here is a link to the appendix that Allen wrote.
Here is the situation in a nutshell. A simplicial set has a small realization. It is a CW complex made of non-degenerate simplices of the simplicial set. The simplices can be glued to themselves or multiply glued to each other. However, the simplicial set structure also implies that the corners of each simplex are consistently locally ordered, and this is not possible with an arbitrary gluing. The consistent local ordering is useful for a variety of purposes. The first time that you'd really care is that it gives you a canonical cup product at the level of simplicial chains.
Geometric topologists, especially 3-manifold topologists, widely use of exactly this structure, except without the consistent local ordering. This is called a generalized triangulation, and you can express it in much the same way as a simplicial set. Instead of using the simplex category, whose morphisms are order-preserving maps between ordered finite sets, you use the symmetric simplex category, whose morphisms are all maps between all finite sets. The small realization of a symmetric simplicial set is exactly a generalized triangulation, if the symmetric simplicial set satisfies a certain freeness condition. The symmetric group $S_n$ acts on the set of formal $n$-simplices, and the condition is that the action on the non-degenerate one should be free. Symmetric simplicial sets appear in a bare handful of papers in the literature.
Every generalized triangulation with consistently locally ordered vertices is represented by a unique simplicial set. This is harmonious view of simplicial sets to make both algebraic and geometric topologists happy. The only problem is that it does not generalize well to other simplicial objects, because the non-degenerate simplices aren't any good in, for instance, a simplicial group.
Simplicial sets are very useful to algebraic topologists. Generalized triangulations are very useful to geometric topologists. Simplicial complexes are useful to combinatorialists: they are hypergraphs with a closure property. Simplicial complexes are not particularly as useful as natural generalizations to either algebraic or geometric topologists, except that they are easy-to-define special cases of : simplicial sets (if you order the order the vertices) ; or generalized triangulations (if you don't order the vertices)vertices. It is true that certain constructions of simplicial set or generalized triangulations are automatically simplicial complexes. For instance the simplicial set of a poset is automatically that (as Charles Rezk says), or the second barycentric subdivision of any type of CW complex that has a barycentric subdivision. (Because the first barycentric subdivision is automatically a simplicial set with colored vertices.)