First of all, loop quantum gravity is a model inconsistent with the existence of the spacetime, gravity, and Lorentz invariance. It doesn't solve any problem with the non-renormalizability of general relativity, either. Instead, the problem of infinitely many counterterms reappears in the infinite number of ambiguities in the "canonical Hamiltonian". See e.g. this most cited loop quantum gravity paper of 2005 and a related paper from 2006:
So we're only talking about string/M-theory here because it's the only known and probably the only mathematically possible consistent quantum theory that includes gravity.
This was a rudimentary correction of a misleading statement in the original question.
Now, to answer the question, general relativity differs from Fermi's theory or a Higgsless theory of massive gauge bosons because in those QFT examples, a consistent short-distance completion that is a local quantum field theory itself exists. In both examples, one finds massive particles at a high scale – the electroweak scale in both cases, in fact – which cure the divergences.
In particular, the seemingly contact (delta-function containing) four-fermion interaction isn't quite a contact one. It comes from the exchange of the W-bosons (and Z-bosons) that are massive. One may say that the original direct, contact, non-renormalizable four-fermion interaction doesn't exist at all; it is an effective description of some deeper, renormalizable interactions coupling the fermions to a gauge boson (cubic vertex).
For a different example, the addition of the Higgs boson cures the problems of the WW scattering but it doesn't change the fact that the W-bosons interact with one another; one only adds an additional term that makes the tree-level amplitudes unitary and that allows the theory to preserve unitarity at higher loops, too.
None of these two scenarios can work in naively quantized general relativity because the source of the non-renormalizability are gravitons, massless particles. They directly interact with each other because this is guaranteed even by the low-energy limit of Einstein's equations which are almost directly implied by the gauge symmetry underlying the metric tensor field (a gauge field of gravity), the diffeomorphism symmetry. Because the basic nonlinearities of the Einstein-Hilbert action induce long-range forces between the gravitons, these interactions can't be due to the exchange of a massive particle. That's why the gravitational interaction can't be generated as an exchange of a massive particle, to emulate the example of the W-boson that is being exchanged to produce the four-fermion interaction.
On the other hand, it's not enough to add some massive matter fields whose exchange cancels the bad behavior of gravity, something that would emulate the "added Higgs to regulate WW scattering" toy model. It's because quantized general relativity produces counterterms that depend on the metric only, like various $R^3$ terms at 2 loops (in the case with no SUSY), and by adding new matter fields, we're just increasing the number of possible types of counterterms depending both on the metric tensor and the matter fields, so these extra matter fields make the situation even worse, not better.
A possible and partial counterexample could be the $N=8$ supergravity which is believed by some people to be perturbatively finite. However, most of the best experts believe that the new counterterms start at the 7-loop level and even if the theory were perturbatively finite (all divergences cancel to all orders), it is clearly non-perturbatively inconsistent (because it doesn't contain the objects charged under the $U(1)$ groups with the correct Dirac quantization condition: one needs the stringy/M completion for that again) and it is phenomenologically unviable because $N=8$ SUSY is too much of a good thing. To consistently break SUSY in $N=8$ SUGRA, one has to apply "stringy" ways to break it, namely by more complicated compactifications of the maximally supersymmetric theories.
One may say the only way to make gravity consistent at high energies is to admit that the graviton is composite. However, quantum field theories don't really allow composite massless particles. The only loophole is string theory which makes graviton a "bound state of string bits", a closed string, and a theory of this kind is exactly the right compromise between the degree of "novelty" that is needed to deal with the harder problem, and the conservativism that is needed not to deviate from the safe consistency waters of quantum field theory. As David Gross would say, string theory is a radically conservative extension of the principles of physics. One may see that despite its not being a quantum field theory in the spacetime in the strict sense, it obeys many conditions and inequalities that may be derived for strictly local quantum field theories and other conditions.
Alternatively, one may consider stringy theories of gravity in various backgrounds, especially AdS-like, to be fully equivalent to a quantum field theory – but one on a spacetime with a different dimensionality. The AdS/CFT is yet another way to see that string theory is radically conservative. Not only string theory avoids radical departures from the rules of QFT that were apparently needed for consistency in the past decades; it is actually fully equivalent to a QFT.
There are many other ways in which string theory is "just a more clever way" to deal with quantum field theories. Matrix theories are QFTs that are also equivalent to sectors of string theory, for a large number of colors, much like AdS/CFT. One may talk about effective field theories and in the perturbative realm of open strings, one may extract all the amplitudes from string field theory, too. String field theory is a rather minor generalization of a quantum field theory although a string field is equivalent to infinitely many fields.
I haven't written the obvious point that gravity corresponds to the dynamics of the whole spacetime which makes things hard by itself. In particular, one cannot have any "universal cutoff scale" that removes excessively high-energy excitations of gravitons. It's because the diffeomorphisms change the proper distances between fixed points in the coordinate space (or spacetime). So many of the procedures to deal with the divergences don't work.
Also, because gravity contains the diffeomorphism group as the gauge group (one needed to remove the negative-norm polarizations of the graviton), general relativity admits no local gauge-invariant operators, see e.g.
Diff(M) as a gauge group and local observables in theories with gravity
That's why one can't talk about the local Green's functions in general relativity, at least not in a formalism that would preserve the Lorentz symmetry. Instead, only the scattering S-matrix may be computed in a manifestly Lorentz-invariant way. That's exactly what string theory does; even if we neglected that gravity shouldn't allow us to calculate the local Green's functions, string theory would force us to realize and learn this fact – not by the general wisdom and thoughts based on field theory but by the cold and indisputable formalism in which string theory just generates the answers.
There are several philosophical attitudes to learn general insights about quantum gravity – some of them build on arguments based on quantum field theory and its decent effects and consistency constraints; others start with very particular, well-defined vacua and calculational schemes in string theory that give us "some examples" what may happen in consistent theories of gravity. At the end, the lessons learned by both of these approaches agree. This agreement has been getting increasingly explicit in the recent decades.
I am afraid that it would make no sense to try to include loop quantum gravity into this discussion because the proponents of loop quantum gravity don't deny just string theory; they deny much of the key material learned in quantum field theory during the last 40 years, too, including the lessons about the renormalization group (especially that it's the infinite ambiguity, and not the "infinities" by themselves, which are the problem), various phases of gauge theory (which have been mapped to the behavior of string theory), the role played by SUSY, monodromies, emergence of extra dimensions in various ways, and so on.
It's really inevitable for the loop quantum gravity proponents to deny most of the modern insights and realizations (both technical and philosophical ones) about quantum field theory because these insights really do imply that approaches such as loop quantum gravity or anything based on "atoms of space" are inevitably inconsistent and have nothing to do with the "real problems". It's really the conservative wisdom about quantum field theory that implies that string theory is the only possible consistent quantum theory of gravity. Thirty years ago, people would be split into various camps (like SUGRA vs string theory) but this ain't the case anymore. People understand that they're investigating many aspects of the same structure from various directions and they understand how some people's findings match other people's findings obtained in a different way so that the whole structure makes sense. It's just a fact that loop quantum gravity or any other research of "atoms of space" fails to be a part of this structure of modern physics.