I think that 秘密 has two usages, one of which is interchangeable with 内緒 and the other is interchangeable with 隠し事 or 秘め事. 隠し事 and 秘め事 are synonyms although 秘め事 sounds more poetic to me.
Cases where 秘密 and 内緒 are correct but 隠し事 and 秘め事 are incorrect:
- That is secret. それは秘密 [内緒] だ。
- Let’s keep this between us. このことは秘密 [内緒] にしておこう。
- secretly 秘密で, 秘密に or 内緒で
In this usage, I feel that 秘密 is more formal than 内緒.
Cases where 秘密, 隠し事 and 秘め事 are correct but 内緒 is incorrect:
“Something which is kept secret” is 秘密, 隠し事 or 秘め事, but not 内緒. For example,
- “He has secret” is 彼には秘密 [隠し事, 秘め事] がある。
As for why 内緒話 is much more common than 秘密話,
it’s secret I do not know. It seems to me like one of the many cases where one phrase is used more often than another for no particular reason. By the way, I would not say 秘密話 is incorrect. If someone uses the word 秘密話, I will understand its meaning naturally and it will not strike me as incorrect.