First of all, as already said by others: Classical class field theory can be formulated entirely without cohomology, so it is a choice to use it.
Benefits of using group cohomology:
If you use Galois cohomology, the main theorems of class field theory can be phrased as statements looking a lot like a form of Poincaré Duality, e.g. local class field theory becomes a nondegenerate duality
$H^i(F,M) \otimes H^{2-i}(F,M^\ast) \rightarrow \mathbb{Q}/\mathbb{Z}$
which looks a bit as if your local field was some kind of 2-dimensional compact manifold (except that the top cohomology group is $\mathbb{Q}/\mathbb{Z}$ instead of $\mathbb{Z}$.
There is a similar reformulation of global class field theory (e.g., look for Artin-Verdier duality).
So, if you like topology, one benefit of choosing to use group cohomology is that you can try to get inspiration/intuition from topology.
Then the Galois group is a little like the fundamental group of "that manifold" and indeed some explicit presentations of the Galois group of a local field look [very vaguely...] a little like presentations of fundamental groups of surfaces. This is a vague analogy at best, but analogies often help us to find things "more natural".
As a side note: Etale cohomology also has a Poincaré Duality theorem, but even though Galois cohomology can be regarded as a special case of étale cohomology, the Poincaré Duality Theorem of étale cohomology says nothing about class field theory. These dualities are disjoint phenomena.
Second: Many things need not be proven afresh. For example, there are maps between group cohomology groups when you go to a smaller or bigger group (restriction and corestriction). From the field perspective this corresponds to going to a bigger field or subfield. If you develop group cohomology, there are automatically induced maps on all group cohomology (or homology) groups. If you avoid group cohomology and instead work with things like the Brauer group, defined through central simple algebras, the respective maps need to be set up manually and you all the time need to verify a lot of little properties which would be automatic if you simply imported them from abstract homological algebra.
Third: Linking back to the first: At some point in your life you might wish to combine number theory with geometry, and consider varieties over number fields. For the varieties you will probably use some cohomology theories of geometric flavour (e.g., etale cohomology of the variety base changed to the separable closure of the base field), or more arithmetic invariants like Chow groups, etc. These invariants call all be phrased cohomologically. For this reason, if you want to combine them with number theory, you benefit a lot if you have also phrased your class field theory in terms of cohomology because then all these concepts "mix in a friendly compatible way". If you avoid using cohomology, you would have to build a lot of connecting bridges beforehand.
Some invariants of arithmetic relevance, e.g., motivic cohomology groups, have been modelled really with topology inspiration in mind (If you read Voevodsky, he clearly thought mostly in terms of homotopy theory). Yet, although topologically inspired they then turned out to be extremely valuable in number theory, too. The same is true for $K$-theory. Many of these things, if one hides their group cohomology origin, might at first appear extemely unmotivated and enigmatic.
