You’re confusing services and their intentions.
KMS provides encryption. Yes it can store small values for you, but that’s it - encrypt this, decrypt that, rotate this key. Everything else is your job. This is the low level service.
Secrets manager builds upon this. More access controls, some built in logic for rotation of secrets, etc.
Which is correct for you depends upon what kind of secret you’re storing, what you do with it, how often and how you need to rotate or regenerate it... dozens of factors.
Basically, KMS is the key master. It manages the keys to the data and will begrudgingly hang on to small bits of paper you’ve asked it to encrypt, but it does nothing else. Data in, encryption, data out, decryption.
If an API key is one of those bits of data that’s fine. It knows not and cares not. Need to rotate it? Pull it out, decrypt it, rotate it, encrypt it, and shove it back in - all that is your business.
Secrets Manager is the gate keeper. It needs the key master but it knows more about who and what is passing through and will help you out from time to time. Stored your most recent allegiance to Zuul but you haven’t reaffirmed it recently? He’ll be a bro and go ahead and send it back in for you and store the new allegiance until you need it again.