Does public relations communication offer a legitimacy that advertising does not have?

Yes. Being in the editorial versus advertising communications of newspapers, broadcasters and web publishers implies a third-party endorsement that's not possible with advertising.

On the other hand advertising offers the kind of control of message that's not possible with public relations.

These concepts can also be applied to paid search advertising versus algorithmic search engine optimization (which is never really "free" because significant human labor is involved).