C++ Exceptions and Inheritance from std::exception

Given this sample code:

#include <iostream>
#include <stdexcept>

class my_exception_t : std::exception
{
public:
    explicit my_exception_t()
    { }

    virtual const char* what() const throw()
    { return "Hello, world!"; }
};

int main()
{
    try
        { throw my_exception_t(); }
    catch (const std::exception& error)
        { std::cerr << "Exception: " << error.what() << std::endl; }
    catch (...)
        { std::cerr << "Exception: unknown" << std::endl; }

    return 0;
}

I get the following output:

Exception: unknown

Yet simply making the inheritance of my_exception_t from std::exception public, I get the following output:

Exception: Hello, world!

Could someone please explain to me why the type of inheritance matters in this case? Bonus points for a reference in the standard.