We have three different things to look at here.
Comprehend Languages says...
For the duration, you understand the literal meaning of any spoken language that you hear.
It goes on to talk about written messages, but this is all we have for spoken language.
You know Druidic, the secret language of druids. You can speak the language and use it to leave hidden messages.
Again, this is all we have for the spoken language. It outlines rules for noticing hidden written druidic messages and specifically mentions magic being useful for understanding hidden writtings... assuming you notice the message to begin with.
In any case, it's pretty clear that Comprehend Languages helps with spoken Druidic. There's no coding or encryption to make what the speaker says carry anything other than "literal meaning".
Druidic is only a "secret" because they don't tell other people about it. However, it's kinda like Fight Club - it's only a secret because the rules make it a secret, security by obscurity. There's nothing beyond the game rules that make it so.
Thieves' Cant says...
During your rogue training you learned thieves’ cant, a secret mix of dialect, jargon, and code that allows you to hide messages in seemingly normal conversation. Only another creature that knows thieves’ cant understands such messages. It takes four times longer to convey such a message than it does to speak the same idea plainly.
Thieves' Cant is different. It isn't really a language at all - it's a system of coding hidden messages inside another language. Nobody speaks Thieves' Cant by itself.
Comprehend Languages specifically says "literal meaning" of the spoken words.
Which means it wouldn't work with Thieves' Cant, as Thieves' Cant is all about coding a secret message within the literal message spoken. The literal meaning and the actual meaning are not the same thing, it's steganography.
Comprehend Languages works on Druidic, but not Thieves' Cant because Thieves' Cant isn't actually a language.