Yes, I'm quite sure that castling rights are the reason for this difference.
But it depends on the engine: some engines may always allow castling (after a position setup) when the king and the rook(s) are still in their original positions, others may not.

As long as your computer is powerful enough to run the emulation with the speed value (x%), you will indeed get the same moves.
You can check this with 'Options > Show FPS' (or key F11) in the engine window: if the value in the upper right corner is about the same as your speed value (x%), then the emulation is correctly running with this speed.