No, a finitely presented simple group does not have any non-trivial finite dimensional matrix representation because every finitely generated matrix group is residually finite. There are many non-isomorphic finitely presented simple groups: Thompson groups $T$ and $V$, Burger-Mozes groups and others. They all have the same fin. dim. unitary representations (trivial).

** Update ** A more interesting question is whether two non-isomorphic residually finite groups can have the same finite dimensional matrix (unitary) representations. The answer, I think, is "yes", and is given by the class of groups, found by Nekrashevych in Trans. Amer. Math. Soc. 362 (2010), no. 1, 389–398. Any two groups in his class have the same finite quotients and every linear representation of any of these groups is finite (the latter result follows from the ``branch" property).