There are multiple explanations of the distinction between the Confederacy of Independent Systems and the Separatist Alliance. They may not all be entirely consistent.
The two terms are somewhat interchangeable
The Databank article suggests that the terms are interchangeable.
The Confederacy of Independent Systems, also known as the Separatist
Alliance, was a collection of star systems and corporate titans that
left the Republic and fought for independence against the Republic’s
clone armies and Jedi Knights. The charismatic former Jedi Count Dooku
was the Alliance’s political leader, while its military power came
from credits and assets donated by powerful galactic trade groups.
There’s also a hint here that the Alliance might refer to the broader collection of CIS plus Trade Federation, etc. In any case, the terms CIS and Separatist Alliance were used both by Separatists and Republic allies, both in the early and later days of the war.
The Alliance predated the CIS
It seems that the Separatist Alliance may have come before the CIS, the former having "congealed" into the latter.
In the blink of an eye, it seemed, the galaxy was embroiled in a
full-scale galactic war. The Separatist Alliance congealed into the
Confederacy of Independent Systems, a coalition of loosely aligned
worlds united for war.
Star Wars Propaganda: A History of Persuasive Art in the Galaxy
The most likely conclusion from this would be that the Separatist Alliance didn’t take on the name “CIS” until there were enough planets and systems involved for there to be a proper government. Dooku’s early efforts were primarily bankrolled by trade and financial groups. There might also be an implication of increasing formalization.
Nor would this be the first name change that the Alliance underwent. This early poster refers to Dooku as the head of “the Independent Movement for Self-Determination.”

Some of it is propaganda
Each side chooses the terminology that makes itself look good, while trying to label the other side as something negative. One person’s rebel is another person’s terrorist.
“Confederacy of Independent Systems” or “CIS” seems to the be the CIS’s own formal and preferred term for itself. All the CIS propaganda in Star Wars Propaganda: A History of Persuasive Art in the Galaxy refers to the Confederacy of Independent Systems, not the Separatist Alliance.

By contrast, officials of the Republic, as well as people not politically affiliated with the Separatist cause, tend to use the term “Separatist.” This is what the pro-Republic artist Janyor of Bith calls it:
I have been a propagandist, a young defender of the Republic, blind to
its faults and gripped by a jingoist’s fervor to support the soldiers
of the Clone Wars. I believed in the rhetoric of Chancellor Palpatine.
I believed in the evils of Count Dooku and the selfish damage
inflicted by the Separatist Alliance.
Star Wars Propaganda: A History of Persuasive Art in the Galaxy
This is also occasionally illustrated in the Clone Wars series:
AHSOKA: Who’s he?
PADME: He is a she, a senator in the Confederacy Of independent systems.
AHSOKA: Your friend is a separatist? One of Dooku’s pawns?
PADME: The separatists used to be a part of the Republic.
The Clone Wars, “Heroes on Both Sides”
The phrase “Confederacy of Independent Systems” seems designed to convey an air of legitimacy, and thus tends to be preferred by the CIS itself. By contrast, “Separatist” is a more neutral or even derogatory term (“the Separatists”).