std::reference_wrapper v.s. int&

I was trying out the std::reference_wrapper with the following snippet

int a = 42, b = 52;
std::tuple<std::reference_wrapper<int>> t = std::make_tuple(std::ref(a));

std::get<0>(t) = b;
std::cout << "t[0] = " << std::get<0>(t) << ", a = " << a << ", b = " << b
          << std::endl;

The output is t[0] = 52, a = 42, b = 52, which not surprising.

However, if I just use auto for t, namely

int a = 42, b = 52;
auto t = std::make_tuple(std::ref(a));

std::get<0>(t) = b;
std::cout << "t[0] = " << std::get<0>(t) << ", a = " << a << ", b = " << b
          << std::endl;

Then I got t[0] = 52, a = 52, b = 52

Looks like the type becomes int&. Then I have some questions:

1. I thought std::ref gives us std::reference_wrapper rather than &?
2. How I should explain the case of & and why it is different from std::reference_wrapper.
3. I also noticed that, in the case of std::reference_wrapper, std::get<0>(t) = 52; does not compile. (While in the case of & it does). the error is "calling a private constructor of class 'std::__1::reference_wrapper'". Could someone explain that in more detail?

Thanks!!