Privacy means that you can talk/act safely in your own closed-off space while no-one knows what you do. The opposite of private is public.
Anonymity means that you can safely talk/act in public space while no-one knows who does it. The opposite of anonymous is ... identified.

If you want your talk be private while doing it in public or via an untrusted service, you can use obfuscation/encryption of the content/payload data of your talk (still anyone could receive it and know it's from you and if they have the key they can decipher it).

If you want to be anonymous in public space, you have to obfuscate the metadata of your talk (so that no-one knows who said it but anyone can still receive it).

*And here is a bit of an overlap depending on where we want to draw the boundary of our privacy realm. In some cases, the knowledge about metadata like location and time of a message can be breach privacy while in other cases this is irrelevant.

You could also do both, meaning you'd have an anonymous appearance in a public/untrusted space, having a conversation with only those people who have the key to your messages. That's a stunt which is not easily accomplished, as obviously you'll need a way to let others know how to reach you, and exchange keys (in other words, you'll have to first make an appointment in private and in a trusted space).

[wanted to write two sentences, no so much text :-D]