A typical example of a closed convex pointed cone S with non-closed S-S is the cone of all non-negative non-decreasing continuous functions in the space C[0,1] of all continuous real-valued functions on [0,1]. The fact that S is closed is evident. On the other hand S - S consists of all continuous functions of bounded variation, so it forms a proper dense subspace of C[0, 1].

The ordering generated by S is a bit funny, but it is an order relation.

thank you Prof. Vladimir Kadets and where I can find a proof of the density of continuous functions of bounded variation in C[0, 1] ?

Every polynomial is a function of bounded variation, and the subset of polynomials is dense in C[0,1]