Model categories provide a powerful framework for commuting (homotopy) limits and colimits, and, more generally, for commuting left adjoint functors and (homotopy) limits, as well as right adjoint functors and (homotopy) colimits. (In what follows, I omit the adjective “homotopy” before limits and colimits.)
In finitely presentable ∞-categories filtered colimits commute with finite limits and in finitely presentable ∞-categories with a set of compact projective generators sifted colimits commute with finite products. However, many other situations of interest are not covered by such statements. For instance, one might want to commute a sifted colimit past an infinite product, a pullback, or a cosifted limit (e.g., a cosimplicial totalization). One might also want to commute sifted colimits past finite products in ∞-categories that do not have a set of compact projective generators, e.g., the ∞-category of small ∞-categories.
In all such situations the relevant statement is false at least for some diagrams, so whatever criterion we devise must analyze the specific diagrams at question. This is precisely what model categories achieve.
For example, one might want to commute K-indexed colimits (for some small diagram K) past some right adjoint functor F: C→B. (For instance, one can take F to be lim_L: Fun(L,B)→B, the limit functor for L-indexed diagrams, which will allows us to commute K-indexed colimits past L-indexed limits.)
We would like to devise a condition on a diagram D: K→C that would guarantee that the canonical comparison map colim_K F(D) → F(colim_K D) is an equivalence in the ∞-category B. This is achieved by the following creative procedure. First, consider the relative ∞-category (i.e., ∞-category equipped with a class of maps (weak equivalences) closed under composition) Fun(K,C) whose weak equivalences are created by the functor colim_K (i.e., a natural transformation of functors K→C is a weak equivalence if its K-colimit is an equivalence in the ∞-category C). The ∞-category Fun(K,B) is turned into a relative ∞-category in the same way. The functor Fun(K,F): Fun(K,C)→Fun(K,B) is a functor between ∞-categories that need not preserve weak equivalences.
Now comes the (potentially) creative part: equip the relative ∞-categories Fun(K,C) and Fun(K,B) with model structures such that Fun(K,F) is a right Quillen functor. In many situations of interest this can be done immediately using existing tools.
Our criterion now says that if D: K→C is a fibrant diagram, then the comparison map colim_K F(D) → F(colim_K D) is an equivalence in the ∞-category B, i.e., the colimit of D over K commutes with F. Different choices of model structures give us different criteria.
For instance, one can take K=Δ^op, B=spaces (alias ∞-groupoids), C=Fun(L,B) (i.e., L-indexed diagrams of spaces) and F=lim_L: Fun(L,B)→B, the L-indexed limit functor, where L can be taken to be the pullback diagram or the infinite discrete category and the model structure can be taken to be the projective model structure. In this case we obtain a criterion for commuting Δ^op-indexed colimits past L-indexed limits. This recovers, for example, the traditional methods for computing homotopy pullbacks of simplicial sets (replace one of the legs by a fibration), infinite homotopy products of simplicial sets (fibrantly replace all terms), etc.
There are many variations on the above theme, for instance, one can consider weighted limits (in the sense of enriched category theory) and then derive the limit functor with respect to both the functor and the weight, which yields even more powerful computational tools etc.
Many classical results on model categories fit in the above framework. For instance, if M is a simplicial model category, X is a cofibrant object, and Y is a fibrant object, then the simplicial mapping space Map(X,Y) from X to Y computes the mapping space in the ∞-localization of M with respect to its weak equivalences. We can fit this in the above framework by taking F=Map(-,Y): Fun(Δ^op,M)→Fun(Δ^op,Spaces), and observing that a cofibrant object X has a cofibrant cosimplicial resolution X⊗Δ^
To answer the original questions:
We need model categories because we need to commute homotopy colimits and limits, more generally, left adjoints and homotopy limits, or right adjoints and homotopy colimits. Such commutation statements are unique to the formalism of model categories and cannot be obtained in other formalisms (without reconstructing a substantial part of the theory of model categories). Another way of saying this is that we want to model operations on objects of interest to us using their “presentations” (alias “resolutions”), which typically are (co)limit diagrams whose homotopy (co)limit is the given object. Model categories explain how operations on resolutions model operations on objects themselves.
Relative categories work just as well for setting up abstract theory, and in fact all formalisms for ∞-categories (e.g., quasicategories, relative categories, complete Segal spaces, etc.) are the same for all practical purposes once the basic theory is set up. (The fact that one hardly ever writes down a quasicategory explicitly is a testament to this claim.) However, in all of the above cases one cannot directly work with the underlying platonic notion of ∞-category (alias (∞,1)-category) and is forced instead to use some presentation (i.e., model, hence the name “model category”). This presentation is nothing else than a colimit diagram of some shape (e.g., a simplicial diagram in the case of Joyal's quasicategories). The desire to manipulate such presentations (i.e., colimit diagrams) efficiently leads us straight to model categories (and the above setup). This explains why all foundational work on these formalisms by Joyal-Lurie, Barwick-Kan, Rezk, etc. uses model categories.
Yes, e.g., in functional analysis (see the work of Costello-Gwilliam, for example). In quantum field theory one has to perform homological algebra with infinite-dimensional vector spaces of smooth functions. Model categories are bound to show up either explicitly or implicitly (for Costello-Gwilliam mostly the latter so far, but this may change soon).