The Moyal and Wick-Voros planes A^{M,V}_{theta} are *-isomorphic. On each of these planes the Poincare group acts as a Hopf algebra symmetry if its coproducts are deformed by twist factors. We show that the *-isomorphism T: A^M_{theta} to A^V_{theta} does not also map the corresponding twists of the Poincare group algebra. The quantum field theories on these planes with twisted Poincare-Hopf symmetries are thus inequivalent. We explicitly verify this result by showing that a non-trivial dependence on the non-commutative parameter is present for the Wick-Voros plane in a self-energy diagram whereas it is known to be absent on the Moyal plane (in the absence of gauge fields). Our results differ from these of (arXiv:0810.2095 [hep-th]) because of differences in the treatments of quantum field theories.
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