There's nothing wrong with having private or protected properties; this is mostly useful when there is some rule or side effect associated with the underlying variable.
The reason why properties seem more natural for public variables is that in the public case, it is a way to hedge one's bet against future implementation changes, whereby the property will remain intact but the implementation details somehow move around (and/or some additional business rule will be needed).
On performance, this is typically insignificant, or indeed identical for straight-assignment properties.
I personally dislike (but often use) plain assignment properties because they just clutter the code. I wish C# would allow for "after the fact refactoring".