A rootkit is the thing you install once you have root - not a way to get root initially. It usually gives the attacker a means to access the machine in the future, even if the vulnerability she used is fixed in the future.
Rootkits are designed to hide themselves. They are essentially attacker installed backdooors.
A backdoor is basically a rootkit that is part of the original software as written by the original developer. The words have different connotations (rootkit is extremely negative, backdoors slightly less).