When should Throwable be used instead of new Exception?

You should not really catch an exception and throw a new one as general as "new Exception".

Instead, if you wish to bubble up the exception just do the following:

try {
    // Do some stuff here
}
catch (DivideByZeroException e) {
    System.out.println("Can't divide by Zero!"); 
} 
catch (IndexOutOfRangeException e) { 
    // catch the exception 
    System.out.println("No matching element found.");
}
catch (Throwable e) {
    throw e; // rethrow the exception/error that occurred
}

It is not good practise, I believe, to catch an exception and throw a new exception instead of the one that was raised to your code block, unless you raise a useful custom exception that provides enough context to elude to the cause of the original exception.

Always throw an Exception (never a Throwable). You generally don't catch Throwable either, but you can. Throwable is the superclass to Exception and Error, so you would catch Throwable if you wanted to not only catch Exceptions but Errors, that's the point in having it. The thing is, Errors are generally things which a normal application wouldn't and shouldn't catch, so just use Exception unless you have a specific reason to use Throwable.

(from comments) The issue that brought this up is that I need to pass an 'exception' to a piece of code a coworker is building if a collection does not get built.

In that case, you might want to throw a checked exception. You could throw an Exception, an appropriate existing subclass of it (except RuntimeException and its subclasses which are unchecked), or a custom subclass of Exception (e.g. "CollectionBuildException"). See the Java Tutorial on Exceptions to get up to speed with Java exceptions.