What separates a public permissioned blockchain from a private one?

Both restrict who can write; they differ in who can read — a public permissioned chain lets anyone read (and thus verify) the data, a private one restricts readers too.

The writer set is gated in both cases (that's what "permissioned" means). The last decision is about readers:

• Public permissioned — anyone may read the chain, so anyone can verify the validity of the stored data. Chosen when public verifiability is required.
• Private permissioned — only a limited set may read. Chosen when the participants don't need outsiders to verify the state and prefer to keep contents confidential.

Note a permissioned chain can be public even though writing is restricted — publicly verifiable content doesn't require publicly writable consensus.