It's interesting to contemplate WHY is there a difference between the two. Security and Private both nasciently have something to do with "info I'd like to keep to myself", but I'm not sure I can come up with a hard delineation between the two, at least when you try to go above "A/S/L" and below "user/pass".

A privacy breach allows... what? Where's a direct threat?

A privacy breach can be a first step to a security breach, though. Social engineering is all about it. Social networks can be, too.

A privacy breach could mean that your job application is rejected, that you pay higher interest on a loan or higher insurance premiums.

It could mean that your health insurance refuses cover for some treatment because some data points towards a pre-existing condition.

You could be rejected as a tenant based on the scoring done by a referencing agency or you could get searched every time you cross a border.

In a more unlikely event could even become the victim of a miscarriage of justice or get blackmailed by a corrupt government official.

Any means of authentication that relies on personal data. SSN, mother's maiden name, postal address, date of birth, etc.