Their philosophies are quite different; Obsidian is much more text and link oriented (it's basically a front-end for markdown), whereas Notion is object-oriented and gives primacy to the kinds of objects that can contain text (blocks, tables, quotes, etc.)

Obsidian can be made to work like Notion and vice-versa, but I see Obsidian as more of a hypertext environment/wiki creator, and Notion as an integrated environment to interface with different ways of interacting with knowledge-focused objects.

I prefer Obsidian largely due to its simplicity, and the fact that it maps really well to how I think. Notion is also incredible for what it does, but it's too much for my brain, and the app is still a bit too slow for the speed with which my thoughts can escape me.

Anytype purports to be like Notion.

Obsidian is plaintext, Notion is richtext. This means, it's simpler to write complex documents in Notion and you don't need to know and handle complicated syntax. Which is good complex stuff like layout, tables, images. Anytype seems to have the same richtext-interface. But with obsidian on the other side, editing can be faster, if you know your syntax, and can type well.

And unlike obsidian, Notion has no direct mechanism for extension, which means there is no way to modify the app, or extend Notion-Documents with new types of data. There is an indirect way, by embedding urls, which comes with some limitations, and ultimatly it means you are depending on other servers, which comes with some more problems.

Not sure how anytype supports extensions.