Probably two different groups
The first group built the androids of Exo III. From "What Are Little Girls Made Of?":
KIRK: Why? Can't your memory banks solve a simple equation like that?
What happened to the old ones, Ruk?
RUK: So long ago.
KIRK: Is it possible they built their machines too well, gave them
pride and a desire to survive? Machines that wanted logic and order
and found that frustrated by the illogical emotional creatures that
RUK: Yes, the old ones. The ones who made us. They grew fearful of us.
They began to turn us off.
KIRK: And isn't it Korby who's creating the same danger to you all
over again? Unlike you, we humans are full of unpredictable emotions
that logic cannot solve.
RUK: Yes. Yes, it had been so long ago, I had forgotten. The old ones
here. The ones who made us, yes. Yes, it is still in my memory banks.
It became necessary to destroy them. You are inconsistent. You cannot
be programmed. You are inferior.
Ruk implies that the Old Ones were destroyed.
By contrast, in "Catspaw" Korob and Sylvia speak of the Old Ones as though they are still alive:
KOROB: We have a duty to the old ones.
SYLVIA: What do they know of sensations? This is a new world.
Since "Catspaw" was broadcast in October 1967 and takes place at Stardate 3018.2, and "What Are Little Girls Made Of" was broadcast in October 1966 and takes place at Stardate 2712.4 , the first episode takes place at a later time, chronologically. If the Old Ones are still alive in that episode, they can hardly be the same group.
Further, Korb and Sylvia are possessed of potent transmutation technology, so far advanced as to seem almost magic. If this is what the Old Ones of that episode provided to them, it seems unlikely that they would have been frightened by mere android consciousness.
So there likely are two groups: one a society of approximately Voyager technology levels that could create sentient AIs of human-like intelligence, the other a species of vastly advanced extra-galactic beings that possessed technology far in advance of the Federation, Borg, or what-have-you.
That said, there is a link between the two episodes.
Both "What Are Little Girls Made Of?" and "Catspaw" were written by Robert Bloch. As suggested on Memory Alpha, Bloch, as a writer of Lovecraftian horror, would have been familiar with the Great Old Ones of Lovecraft's mythos, which might have inspired him to include the term in the episodes he wrote. Indeed, the Lovecraftian Old Ones could be the same as the entities in "Catspaw." Still, the dissimilarity between the two groups of Old Ones in the episodes written by Bloch is sufficiently great that they are very likely distinct.