(Skip to the end for the actual question.)
Definitions
According to Oxford Living Dictionaries, an initialism is:
"An abbreviation consisting of initial letters pronounced separately".
According to the OED's definition of acronyms, initialisms are also acronyms:
"A group of initial letters used as an abbreviation for a name or expression, each letter or part being pronounced separately; an initialism".
General Examples
Many of those initialisms are used with the article 'the', some examples (using the abbreviation and writing the word or expression in its full form):
I live in the US. <=> I live in the United States.
The CIA operates globally. <=> The Central Intelligence Agency operates globally.
General 'rule' on the subject
I agree with this answer, which establishes that generally speaking, initialisms should be used with the word 'the':
"Whenever its full form (Federal Bureau of Investigation) is normally preceded by "the," I would expect its initialized form to be as well. "
Exceptions to the rule
I have, however, found that this is not always the case, for example with the initialism CDC (Centers for Disease Control and Prevention, a public health institute in the US).
Another exception is 'ATF' (Bureau of Alcohol, Tobacco, Firearms and Explosives, US law enforcement agency), which is mentioned briefly at the very end of the post.
First of all, we need to establish that the full form is commonly (perhaps not exclusively) written with 'the':
This seems to be the case in academia, 700 thousand+ results on Google Scholar
It also seems to be the case in the media: 160 thousand+ results on news.google.co.uk
Now some examples in which the initialism CDC is used without a definite article:
Some quotes from the CDC's own website:
"CDC is one organization"
"hard work make CDC a better organization"
"Investing and acting globally enables CDC to be better prepared"
"Progress has been made in the year since CDC first responded to the Ebola outbreak"
"Recognizing that CDC’s nearly 30 years of collaboration"
"the cornerstone of CDC’s mission of service "
"Today, CDC is one of the major operating components of"
"The CDC" is also used by many (where CDC is used as a noun):
There are 200 thousand+ results on news.google.co.uk.
There are almost half a million results in academia, according to Google Scholar.
My question
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?
Additionally, 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?
*For completeness, a list of conventions
Initialisms which are almost without exception used with the:
FBI, CIA, BBC, ATM
Initialisms which are almost without exception used without the:
CERN, POTUS, FLOTUS, NATO, NASA, TLS, SSL
Initialisms which are sometimes used with, and sometimes without the:
CDC (see examples above), ATF (without, see first paragraph, this NYT article actually uses A.T.F. with and without the)
Best Answer
As a native speaker, I think I can answer both your questions!
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.
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.