The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$, with respect to the difference operator $\phi f(z)=f(qz)$, $2\le q\in\mathbb{Z}$, arising from an endomorphism of the elliptic curve. Our main theorem says that such an $f$ satisfies, in addition, a polynomial differential equation with coefficients from $K,$ if and only if it belongs to the ring $S=K[z,z^{-1},\zeta(z,\Lambda)]$ generated over $K$ by $z,z^{-1}$ and the Weierstrass $\zeta$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-difference operators or Mahler operators). While the general approach, of using parametrized Picard-Vessiot theory, is similar, many features, and in particular the emergence of monodromy considerations and the ring $S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over elliptic curves which is a genus one analogue of the integrability results obtained by Sch\"afke and Singer over the projective line.