There are three mentions of "secret fire" in the Silmarillion.
- ...and Ilúvatar shall give to their thoughts the secret fire, being well pleased.
- ...and the Secret Fire was sent to burn at the heart of the World...
- ...and Fëanor grew swiftly, as if a secret fire were kindled within him.
Of these the first two are evidently referring to the same thing as the Flame Imperishable, whereas the third is just a generic reference to "a fire that is secret", and so may be discounted.
If we cross-reference these with the source texts given in History of Middle-earth 10 and 5, we find the following:
- This is also "secret fire" in History of Middle-earth 10 but is "secret Fire" in History of Middle-earth 5.
- This is also "Secret Fire" in History of Middle-earth 10 but is likewise "secret Fire" in History of Middle-earth 5.
So we can see that the capitalization was consistent in the original texts from the 1930s, but was changed in the 1950s revsions, with the first going to all lowercase but the second going to all title-case.
Christopher Tolkien does not account for this change and this difference in his commentary.
The other notable mention of "secret fire" is, of course:
I am a servant of the Secret Fire, wielder of the flame of Anor. You cannot pass.
And again we see that its title-case too.
In the end we have no explanation for this change. We can say for certain that it's not a publishing error as it's consistent with the latest source texts from History of Middle-earth 10, and it seems highly unlikely to be a simple authorial error as Christopher Tolkien maintained it from the source texts to the published Silmarillion.
Ultimately we just don't know.