I've heard the term "backdoor" used for a long time before "rootkit" or "advanced persistent threat", so it may be a generational thing.
From https://en.wikipedia.org/wiki/Rootkit:
> The modified compiler would detect attempts to compile the Unix login command and generate altered code that would accept not only the user's correct password, but an additional "backdoor" password known to the attacker...This exploit was equivalent to a rootkit.
From https://en.wikipedia.org/wiki/Advanced_persistent_threat:
> Establish Foothold – plant remote administration software in victim's network, create net backdoors and tunnels allowing stealth access to its infrastructure.
From 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.
I read all of that as a backdoor being an umbrella term, of which one type is a rootkit, and APTs create backdoors, perhaps of a type other than rootkit (e.g. net backdoor).