Most fundamentally, I think, including the word 'diplomacy' implies a certain semblance of legitimacy.
The EU and NATO use the term 'public diplomacy' to refer to their communications with publics of member states and (in NATO's case) publics in whose countries they are engaged in operations. This is because talking of 'diplomacy' rather than 'affairs' implies that they are legitimate international actors, on a similar level to states.
Within the United States' political discourse 'diplomacy' is usually referred to as being in opposition to militarism thus it is generally seen as the peaceful, liberal option. Talking of public 'diplomacy' then also invokes this implication. Any historian of diplomacy will tell you how weird this interpretation is (one neat aphorism among many on diplomacy is that it is the art of 'saying good doggie until you can find a rock') but that's how its generally talked about nevertheless.
In short, public diplomacy is a phrase used when people want to legitimise and give a liberal sheen to their practices - public affairs is when they don't feel the need to do that so much.
Ultimately PD is whatever anyone says it is. Its most widely held definition is that spoken by the loudest voice - simple as that.
Or as Nietzsche said, "only that which has no history is definable"