Very good question. Wormholes open up to another universe ( or a far away part of our universe) on the other side. Blackholes do not.
But blackholes are generally agreed to be exact solutions to general relativity (GR) using only ordinary matter to form. Wormholes are more controversial, and require exotic matter, or something else unusual in order to form and not collapse, according to GR.
To expand on the above response, note that there are several different types of wormholes. The main (and most easily understood) distinction to draw is between traversable and nontraversable wormholes. Nontraversable wormholes have the property that although they mathematically bridge to distinct regions of spacetime (the case in which the wormhole connects two regions of the same spacetime is slightly fuzzier), it is impossible for any signal, be it a person or a beam of light, to reach the other side without smashing into a singularity. These types of "wormholes" are actually coincident with black holes, so there is no distinction between these two.
This is not generally what the layperson thinks of as a wormhole, then, as most people want something they can jump through and emerge from unscathed. These traversable wormholes are quite different from black holes in the sense that they are completely different solutions to Einstein's equations. As jimgraber notes, though, exotic matter is generally necessary to arrive at one of these traversable wormhole situations, so at the moment their theoretical basis rests on the plausibility of such a quantity of exotic matter existing.
In case you are unfamiliar, exotic matter is a class of matter which either a) has imaginary mass or b) has negative mass. This is a distinct concept from antimatter, which still has positive mass and obeys all known laws of physics.
Is the distinction clear (both between exotic matter and antimatter and black holes and worm holes)?
A singularity is, in general, a point at which an equation gives a nonsensical answer, either infinity or something divided by zero. In the context of this discussion, this singularity is where the gravitational forces (spacetime curvature) go to infinity.