I tend to add lots of assertions to my C++ code to make debugging easier without affecting the performance of release builds. Now, assert is a pure C macro desi
Assertions are entirely appropriate in C++ code. Exceptions and other error handling mechanisms aren't really intended for the same thing as assertions.
Error handling is for when there's a potential for recovering or reporting an error nicely to the user. For example if there's an error trying to read an input file you may want to do something about that. Errors could result from bugs, but they could also simply be the appropriate output for a given input.
Assertions are for things like checking that an API's requirements are met when the API wouldn't normally be checked, or for checking things the developer believes he's guaranteed by construction. For example if an algorithm requires sorted input you wouldn't normally check that, but you might have an assertion to check it so that debug builds flag that kind of bug. An assertion should always indicate an incorrectly operating program.
If you're writing a program where an unclean shutdown could cause a problem then you may want to avoid assertions. Undefined behavior strictly in terms of the C++ language doesn't qualify as such a problem here, since hitting an assertion is probably already the result of undefined behavior, or the violation of some other requirement which could prevent some clean-up from working properly.
Also if you implement assertions in terms of an exception then it could potentially be caught and 'handled' even though this contradicts the very purpose of the assertion.