I would like to add some more details on Achim's answer. One of the most important applications of the Dold-Kan correspondence is to help us define the derived functors of non-additive functors. Let's come back to the classical construction:

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$ be an **additive** functor between abelian categories. Suppose that $\mathcal{A}$ has enough projectives then for every $A$ we can take a quasi-isomorphism $\mathbf{P} \overset{\sim}{\longrightarrow} A$ then define the left-derived functors by
$$L_i F(A) =  H_i(F\mathbf{P}) \ \forall \ i \geq 0.$$ What are the advantages of being additive in this definition?

 - First we have used $F(d) \circ F(d) = F(d^2) = F(0) = 0$ (being a functor and $F(0)=0$) to show that $F\mathbf{P}$ is indeed a complex.
 - Second we want out derived functors defined uniquely up to isomorphisms. What we usually do in homological algebra is the **comparison lemma** which basically says that projective resolutions is unique in some sense (up to a chain homotopy). The proof of this lemma combines these following (just look at any book on homological algebra)
$$f - g = ds + sd \ \ \text{and} \ \ \text{additivity of} \ F \Rightarrow F(f) - F(g) = F(d)F(s) + F(s)F(d).$$ 

In a word, additive functors preserve homotopies of chain complexes, and what we have done so far is taking the composition of the following (where $\mathbf{Ch}_{\geq 0}(F)$ is applying $F$ degree by degree, and only well-defined with $F$ additive)
$$A \in \mathcal{A} \overset{\mathbf{P}}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{A}) \overset{\mathbf{Ch}_{\geq 0}(F)}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{B}) \overset{\text{homology}}{\longrightarrow} L_i F(A) \in \mathcal{B}.$$ Now if we lose the additivity of $F$, we must go other ways around, perhaps the most natural way is via the Dold-Kan correspondence (at least the category of simplicial sets - or abelian ones if you like - is not different that much from the category of chain complexes at least from the point of view of model categories theory). Let $N: \mathbf{Ch}_{\geq 0}(\mathcal{A}) \longrightarrow s\mathcal{A}$ (the right side is the category of simplicial objects on $\mathcal{A}$) be one-side equivalence in the Dold-Kan correspondence. Then we can go as in the following diagram
$$A \in \mathcal{A} \overset{\mathbf{P}}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{A}) \overset{N}{\longrightarrow} s\mathcal{A} \overset{sF}{\longrightarrow} s\mathcal{B} \overset{N^{-1}}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{B}) \overset{\text{homology}}{\longrightarrow} L_i F(A) \in \mathcal{B}.$$ In this way, we can always talk about derived functor, say of an "arbitrary" $T$, but one should be careful since this new definition is a bi-degree one $L_i T(-,n)$. When $T$ is additive, one has
$$L_i T(-,n) \simeq L_{i-n}(T).$$
Two of the most important class of non-additive functors are the class of symmetric functors $\mathrm{Sym}$ and the class of free group-algebra functors $\mathbb{Z}$. Derived functors of these two classes coincide and give us the homology groups of Eilenberg-MacLane spaces. So at least theoretically, one knows why the (co)homology of Eilenbeg-MacLane spaces are legendarily difficult because they are derived functors of **non-additive** functors. Luckily, a full-fledged computation of these groups was done by H. Cartan in his famous *Séminaire Henri Cartan* (1954-1955). For more details on my comments, you can consults works of Dold and Puppe.