At a very concrete level, Turaev-Viro invariants of a compact 3-manifold (with or without boundary) can be easily computed by a computer from a triangulation and very often (although not always) distinguish non-homeomorphic manifolds.
To calculate a Turaev-Viro invariant you need to fix a level $r=3,4,\ldots$: for $r=5, 7$ you already obtain a quite powerful (and mysterious) invariant, which works on any kind of compact 3-manifold. For instance, it helped to distinguish immediately most of the non-homeomorphic manifolds in these lists.
So, distinguishing many triangulated 3-manifold 3-manifolds is maybe "an example where using classical theories is hard, but using a tqft is comparatively easy". The "classical theory" here would involve recognizing prime summands, decomposing along tori, finding a hyperbolic structure, etc. etc.
Note however that the cost of calculating Turaev-Viro invariants increases exponentially with $r$ and the number of tetrahedra, so I don't know if they can be effectively used to distinguish -- say -- two manifolds having 20 tetrahedra.
