I know both of them, albeit at different levels of detail. But we can start with the differences I came across, perhaps more of them :-).
Scale
First, the Quantum platform provides a complete working environment for UML state machines, while boost :: statechart only helps in the implementation of the state machine. Thus, boost :: statechart provides the same mechanism as the Quantum Platform Event Processor (QEP).
UML Compliance
Both approaches are designed to conform to UML. However, the Quantum platform performs transition actions before the exit actions of the corresponding state. This is contrary to UML, but in practice this is rarely a problem (if the developer knows this).
Boost :: statechart is designed in accordance with UML 1.4, but as far as I know, the execution semantics have not changed in UML 2.0 in an incompatible way (please correct me if I am wrong), so this should not be a problem.
Supported UML Features
Both implementations do not support the full set of UML state machine functions. For example, parallel states (aka AND) are not supported directly in QP. They must be done manually by the user. Boost :: statechart does not support internal transitions, as they were introduced in UML 2.0.
I think the exact features supported by each technique are easy to find in the documentation, so I don't list them here.
In fact, both support the most important statechart functions.
Other differences
Another difference is that QP is suitable for embedded applications, while boost :: statechart may not be possible. The FAQ says “it depends” (see http://www.boost.org/doc/libs/1_44_0/libs/statechart/doc/faq.html#EmbeddedApplications ), but for me this is already a big warning sign.
In addition, you must perform special measurements to get boost :: statechart in real time (see frequently asked questions).
So many differences that I know, tell me, if you find more, I would be interested!