What is the difference between the two? Is it just a renaming?
I don't pretend to be an expert in the field, but from what I understand of it the National Guard are completely new and answerable only to the presidency. Unlike other state organs, like the Ministry of Internal Affairs; who have their own hierarchies to answer to. Officially the National Guard's purpose is counter-terrorism, which is why they have such great legal and military powers.
But it has been suggested that this is because Putin is attempting to counter-balance the power of other institutions. By making the National Guard a military force with generous legal powers they will have authority above police, defence, FSB, and internal/emergency forces. The National Guard's commander is Viktor Zolotov; Putin's long time bodyguard, meaning they are effectively Putin's own army with the right to arrest and detain whomever they like without reason for as long as they like. This might be Putin's way of ensuring that any other eligible presidential candidates cannot use state organs against him.