What is a topological feature, that a (some) tqft (e.g. in 3 or 4 dim) sees but homology/cohomology/homotopy groups dont? Or: what is an example where using classical theories is hard, but using a tqft is comparatively easy?
All the answers so far have focused on 3 dimensions, but the answer is much more striking in 4 dimensions. Freedman's theorem tells you that classical homology invariants give you complete information about topological, simplyconnected 4manifolds. These classical invariants cannot, however, distinguish between distinct smooth structures on the same topological 4manifold, and essentially our only technique for distinguishing smooth 4manifolds is Donaldson's invariant or the SeibergWitten invariant or their relatives. These do not quite form a TQFT, but are related to TQFTs. Edit: On request, a little about how the 4manifold invariants are related to a TQFT. This is all nicely explained in the beginning of Kronheimer and Mrowka's book Monopoles and 3manifolds. There are actually three different theories, denoted $\widehat{\mathit{HM}}$ ("HMfrom"), $\check{\mathit{HM}}$ ("HMto", unfortunately typeset badly here), and $\overline{\mathit{HM}}$. All are close to satisfying axioms for a TQFT assigning a vector space to a 3manifold and maps to a 4manifold, at least for connected manifolds. (The vector spaces are infinite dimensional, but finite in each graded piece.) Unfortunately, however you slice it, in each case the invariant associated to a closed 4manifold in the usual TQFT way (when defined) is zero. Instead, you use the fact that there is an exact triangle $$ \cdots \longrightarrow \widehat{\mathit{HM}} \longrightarrow \overline{\mathit{HM}} \longrightarrow \check{\mathit{HM}}\longrightarrow \cdots $$ (with right mapping to left), and the map $\overline{\mathit{HM}}(W)$ is $0$ for $b_2^+(W) \ge 1$. If you have a 4manifold $W$ with $b_2^+(W) \ge 2$, you factor it as two cobordisms $W = W_1 \cup_Y W_2$ for some 3manifold $Y$, with $b_2^+(W_i) \ge 1$. Then the properties above let you map from $\check{\mathit{HM}}(S^3)$, to $\check{\mathit{HM}}(Y)$, backwards in the exact triangle to $\widehat{\mathit{HM}}(Y)$, and then forwards to $\widehat{\mathit{HM}}(S^3)$. The resulting map (from $\check{\mathit{HM}}(S^3)$ to $\widehat{\mathit{HM}}(S^3)$) gives the interesting SeibergWitten invariants of $W$. 


The only topological information in 3manifolds besides homology and homotopy is Reidemeister torsion (see this question). TQFT sees RaySinger torsion, which is the same thing. Indeed, it was this discovery by Schwartz (and independently, unpublished, by Singer) 


Rasmussen's $s$ invariant detects nonsliceness of some knots that no other method applies to. 


The Tait flyping conjecture was proven by Menasco and Thistlethwaite using knot polynomial invariants (which are a version of TQFT invariants). There is a lower bound on the braid index of a knot in terms of the Jones polynomial. I don't think that there is an efficient algorithm to compute the braid index of knots in general using geometric techniques, so sometimes this works better. There are related estimates of tunnel number and Heegaard genus in terms of TQFT invariants, but these are not sharp in many cases. However, computing TQFT invariants is straightforward, but exponential, so I'm not sure these estimates are necessarily "easier". Estimates of Heegaard genus for Seifert spaces were given Boileau and Zieschang using algebraic techniques, and this has been done by Helen Wong using TQFT invariants. 


At a very concrete level, TuraevViro invariants of a compact 3manifold (with or without boundary) can be easily computed by a computer from a triangulation and very often (although not always) distinguish nonhomeomorphic manifolds. To calculate a TuraevViro 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 3manifold. For instance, it helped to distinguish immediately most of the nonhomeomorphic manifolds in these lists. So, distinguishing many triangulated 3manifolds 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 TuraevViro 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. 

