Control difference between Hide() and Visible?

They are equivalent. From the documentation for Control.Hide:

Hiding the control is equivalent to
setting the Visible property to false.

You can confirm this with reflector:

public void Hide()
{
    this.Visible = false;
}

You might use Show() or Hide() when you know the value and use Visible when you take the visibility in as a parameter, although personally I would always use Visible.