Some history: In the pre-Wilsonian interpretation of quantum field theory, renormalisability of theories was considered an essential requirement. That is, you should be able to remove all ultraviolet divergences of the theory (that occur in perturbation theory) by absorbing them in a finite number of parameters in the lagrangian. If you could not do that, then the theory was called "non-renormalisable" and considered pathological. The renormalisability criteria placed strong constraints of the theory, determining which finite number of interactions were allowed. The standard model is renormalisable.
One effect of the renormalization process was that coupling constants became scale (energy) dependent, and such running coupling constants were understood to be physical. eg asymptotic freedom is the result of the QCD coupling constant becoming weak at large energies.
Wilson's goal and approach in condensed matter was different. He was interested in studying the behaviour of complicated theories near the second order phase transition where the correlation length becomes very large, that is when the theory is essentially scale invariant. That means that you could study the system via an effective theory near the phase transition point without worrying about the underlying microscopic degrees of freedom. In this approach you write down a simple field theory with the desired symmetries and you use a cut-off $\Lambda$ in all calculations since your theory is only an approximation at low energies $E \ll \Lambda$ (long distance scales).
So in the Wilsonian approach there are no ultraviolet divergences. But it also meant that you had to include many more interactions in your lagrangian. The coupling constants also became $\Lambda$ dependent and you had a running of the couplings here too. In practice, in the limit $E/\Lambda \to 0$, only a finite number of couplings are important: the ones that are "marginal" and "relevant". The ones that are "irrelevant" are the ones that a particle physicist would have labelled as "non-renormalisable".
So it was initially a different philosophy and approach in high-energy physics and condensed matter, but gradually high-energy physicists understood that the Wilsonian perspective of effective field theories is applicable to their studies of quantum field theory.
In the modern perspective, the standard model is believed to be a low-energy approximation of some as yet unknown theory. Although the current model is renormalisable, it is believed that "non-renormaliable = irrelevant" interactions would become important at higher energies, and they would represent new processes. So you can write down what the next possible interaction could be (eg to give neutrinos small masses) and make some predictions.
Such an effective field theory approach is also useful when you want to make low energy predictions from your high energy theories.
So, in summary, the Wilsonian perspective of quantum field theories and the renormalisation group is important because it gives them a very physical and intuitive meaning.
Now, some comments about "conformal field theories". A bigger symmetry than just scale invariance is "conformal invariance". In two dimensional space conformal invariance results in an infinite dimensional symmetry group which places strong constraints on a theory and allows many exact results to be obtained in condensed matter systems. It is also important in string theory as the world sheet is two dimensional. Conformal symmetry is less powerful in higher dimensions as the symmetry group is finite and the symmetry is typically broken by quantum effects through the generation of mass scales.