(Replying to PARENT post)
https://en.wikipedia.org/wiki/Rootkit
"A rootkit is a collection of computer software, typically malicious, designed to enable access to a computer or areas of its software that would not otherwise be allowed (for example, to an unauthorized user) while at the same time masking its existence or the existence of other software."
https://en.wikipedia.org/wiki/Backdoor_(computing)
"A backdoor is a method, often secret, of bypassing normal authentication in a product, computer system, cryptosystem or algorithm etc. Backdoors are often used for securing unauthorized remote access to a computer, or obtaining access to plaintext in cryptographic systems.
A backdoor may take the form of a hidden part of a program,[1] a separate program (e.g. Back Orifice may subvert the system through a rootkit), or may be a hardware feature.[2] Although normally surreptitiously installed, in some cases backdoors are deliberate and widely known. These kinds of backdoors might have "legitimate" uses such as providing the manufacturer with a way to restore user passwords."
(Replying to PARENT post)
The connotation difference is the difference between getting hit with a 10mm and a 9mm. Negligible, as it's leaving a hole that you really don't want there.
(Replying to PARENT post)
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).