This is slightly simpler than the arguments above. The double covers of (0,1) surgeries for these two manifolds have different homologies.
The double cover of the (0,1) surgery on the figure 8 knot complement has homology Z/5Z + Z. The double cover of the (0,1) surgery on the trefoil knot complement has homology Z/3Z + Z.
It is unclear how well distinguishing manifolds by the homologies of their covers works and it is known to fail for certain Sol torus bundles. However, when it succeeds, it provides a concrete invariant for distinguishing manifolds.
Also, the double cover of any two bridge knot complement S^3\TBL(p,q) is the complement of a null-homologous knot complement in L(p,q). Hence it has homology Z/pZ + Z as does the double cover of (0,1) surgery (I am implicitly using that the (0,1) curve lifts in cyclic covers). In this case, the trefoil is the (3,2) two bridge knot and the figure eight knot is the (5,2) two bridge knot. So this is an effective technique for distinguishing many pairs of (0,1) surgeries on two-bridge knot complements.