Assertions can be left on simply by not defining NDEBUG, so that's not really an issue.
The real problem is that assertions call abort(), which instantly stops the program. This can cause problems if there is critical cleanup your program must do before it quits. Exceptions have the advantage that destructors are properly called, even if the exception is never caught.
As a result, in a case where cleanup really matters, exceptions are more appropriate. Otherwise, assertions are just fine.
