As a native speaker, I think I can answer both your questions!
Is there a reason why both versions are used with CDC, whereas most other initialisms seem to be used either with, or without, but not both?
In speaking and writing, a the is generally put before initialisms, which are spelled out when spoken: both for organizations like the FBI (eff-bee-ahy) and the NSA (en-ess-ay) and for countries like the US (you-ess) and the UK (you-kay).
We rarely, if ever, put a the before acronyms, which are spoken as a word: from NASA (nah-suh) to FLOTUS (flow-tuss) and POTUS (poh-tuss), they're all the-less.
Are both versions correct? That is, can CDC and the CDC be used? If so, are they interchangeable or do both have their own use case?
Generally, I only use or hear the CDC and the ATF, as befitting the usual American English pattern. Given the CDC's zealous use of just "CDC" on their own website, however, it seems they might have done away with the the themselves—possibly to seem more friendly given their "I am CDC" video.
However, as is typical with English, one says whatever sounds right and there's always flexibility and change over time.
Best Answer
There doesn't seem to be an official term for words made by spelling out the name sounds of an initialism's letters. Twelve years ago today, on 2000-01-06, Lee Daniel Quinn suggested calling this type of word a vocologue, but his attempted neologism doesn't seem to have caught on.
I'd like to see a happier fate for wim 9's acronomatopoeia, suggested in the comments above. I like the cleverness of the portmanteau formation, and it can be reasonably-well understood to mean a word formed by the sounds of an initialism's letter names.
I was able to find only the following 9 common examples of acronomatopoeia: