They're not necessarily equivalent as tensor categories. 

However, there are examples of finite groups (smallest of order 64) with representation categories which are equivalent as tensor categories but not as symmetric tensor categories (see e.g. [arXiv:math/0007196](https://arxiv.org/abs/math/0007196 "Pavel Etingof, Shlomo Gelaki: Isocategorical groups")). In other words, in some cases the same abstract tensor category might be endowed with inequivalent symmetric structures (you can think of these as the pullback of the standard symmetry of the category of vector spaces through inequivalent embedding functors).