When debugging parsers for my toy programming languages print debugging is less helpful and I make heavy use of all the debug tools you mention. The same goes for most types of business logic—writing a test and stepping through it in the debugger is usually the way to go.
But when troubleshooting odd behavior in a complex web app, the inverse is true—there are usually many possible points where the failure could occur, and many layers of function calls and API calls to check, which means that sticking a debug statement prematurely slows down your troubleshooting a lot. It's better to sprinkle logs everywhere, trigger the unexpected behavior, and then skim the logs to see where things stop making sense.
In general I think there are two conditions that make the difference between the debugger or print being better:
* Do you already know which unit is failing?
* Is there concurrency involved?
If you don't yet know the failing unit and/or the failing part of the code is concurrent, the debugger will not help you as much as logs will. You could use logs to narrow down the surface area until you know where the failure is and you've eliminated concurrency, but you shouldn't jump straight to the debugger.