Dear sb1, the top experts behind the Consistent Histories often credit Hugh Everett - the father of the many-worlds interpretation - with making the foundational quantum physics community focus on the "histories" as the basic entity whose probability quantum mechanics predicts. However, this is where the overlap ends.
People behind Consistent Histories usually admit that their interpretation - my favorite one - is just a refinement of the probabilistic Copenhagen interpretation. Nothing essential has changed; the predictions are still fundamentally probabilistic. Consistent Histories is the framework that incorporated the explanations of decoherence - the key process that calculates the boundary of the classical and quantum world - as the first one (and maybe still only one). Many-worlds interpretation is just a semi-popular psychological framework to think about quantum mechanics - and it hasn't been useful to do any actual, new calculations. One doesn't really know how to extract the numerical values of the probabilities from the many worlds, at least not in a way that would tell us more than any other interpretations.
Niels Bohr is said to have understood the essence of decoherence but he was unable to transform his understanding into comprehensible explanations or equations.
Decoherence is not quite the collapse of a wave function
Also, you say that there is a similarity between decoherence and the collapse of the wave function. Indeed, decoherence is what replaces the flawed ideas about the "collapse" and it is the key process that creates "classical objects" for which our non-quantum intuition is OK. However, all the details are completely different.
First of all, decoherence can be fully derived from the standard equations of quantum mechanics - from evolution combined with tracing over the environmental degrees of freedom. It doesn't try to distort the laws of quantum mechanics in any way: it just shows a generic and previously unappreciated consequence of these equations that is responsible for explaining lots of old puzzles.
On the contrary, the idea of a "collapse of the wave function" always tries to incorrectly treat the wave function as a real classical wave. For it to "physically" collapse, there must be new mechanisms and new terms added to the equations that make it collapse at a certain point. Needless to say, all these mechanisms are fictitious and all the mechanisms responsible for such a huge effect can be easily excluded. Quantum mechanics doesn't imply anything of the sort and the tests of quantum mechanics are not compatible with anything of the sort.
An important technical difference between decoherence and the collapse of the wave function is that decoherence actually doesn't decide which result will be measured. It just diagonalizes the density matrix to $\mbox{diag}(p_1,p_2,\dots,p_N)$ where $p_i$ then play the role of the probabilities that the individual preferred basis vectors will be detected. Because the information about the relative phases is getting quickly lost, the off-diagonal elements rapidly converge to zero. But decoherence never transforms the density matrix with the many $p$'s to something like $(0,0,1,0,\dots,0)$. Never, ever. Quantum mechanics remains the only player here and its predictions remain and will always stay probabilistic. One can never and one will never restore determinism and decoherence doesn't try to do anything of the sort.
