Do we still need model categories? One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak equivalences; better, if you have a simplicial model category, is to take the homotopy coherent nerve of the fibrant-cofibrant objects). 
What other functions, then, do  model categories serve today? I understand that getting the theory of $\infty$-categories off the ground (as in HTT, for instance) requires a significant use of a plethora of model structures. However, if we assume that there exists a good model of $(\infty, 1)$-categories that satisfies the properties we want (e.g. that mapping sets are replaced with mapping spaces, limits and colimits are determined by homotopy limits of spaces), how are model categories useful?
I suppose one example would be computing hom-spaces: a simplicial model category gives you a nice way of finding the mapping space between two objects in the associated localization. However, in practice one only considers cofibrant or fibrant objects in the $\infty$-category in the first place, as in Lurie's construction of the derived $\infty$-category (basically, one considers the category of projective complexes -- for the bounded-above case, anyway -- and makes that into a simplicial category, and then takes the homotopy coherent nerve). 
One example where having a model structure seems to buy something is the theorem that $E_\infty$ ring spectra can be modeled by 1-categorical commutative algebras in an appropriate monoidal model category of spectra (in DAG 2 there is a general result to this effect), and that you can straighten things out to avoid coherence homotopies. I don't really know anything about $E_\infty$-ring spectra, but I'm not sure how helpful this is when one has a good theory of monoidal objects in $\infty$-categories.
 A: Here are some rough analogies:


*

*Model Category :: $(\infty, 1)$-category

*Basis :: Vector space 

*Local coordinates :: Manifold


I especially like the last one. When you do, say, differential geometry in a coordinate free way you often end up with more conceptual, cleaner, and more beautiful proofs. But sometimes you just have to roll up your sleeves, get your hands dirty, and compute in local coordinates. Sometimes different local coordinates are useful for different calculations.  
One modern point of view is that Quillen Model categories are the local coordinates of a homotopy theory (i.e. presentable $(\infty,1)$-category, c.f. this MO answer). When you can compute without choosing local coordinate (a model category structure), that is great. But sometimes it can be very helpful (maybe even necessary) to pick a Model structure to do a specific computation.  Sometimes it is helpful to jump between different Model structures which describe the same underlying $(\infty,1)$-category. 
Model structures are still extremely useful, even essential to the general theory. But they are a tool, just like local coordinates in differential geometry. 
A: One nice feature of model categories is that you can speak also of the non-bifibrant objects (which is not longer possible, once you passed to the corresponding infinity-category). A few examples where this is useful:


*

*Simplicial sets: non-Kan simplicial sets appear again and again. For example, the n-simplex itself.

*Spectra: in most models for spectra all fibrant objects are $\Omega$-spectra. One often wants to consider non-$\Omega$-spectra like Thom spectra or suspension spectra.

*Diagram categories: The process of replacing a map (say, between topological spaces) by a fibration can be seen as a fibrant replacement in the arrow category. 

*Chain complexes: not all modules are projective...


For proving abstract theorems, the framework of $\infty$-categories seems to be in many senses very convenient. But model categories are (in my opinion) often nicer if you want to deal with concrete examples (which are often not bifibrant) and want to see how to compute derived functors of them. Also, concrete models of spectra (like symmetric spectra) where $E_\infty$-rings are modeled by strictly commutative monoids are really nice to write down concrete examples.
A: I confess to being confused by all this $(\infty,1)$ category business, and the way $\Pi X$ is used as another name for the singular simplicial set of $X$. This is related to Peter's question on computations. 
I thought one reason for moving from loops or paths to fundamental groups or fundamental groupoids, i.e. taking homotopy classes, was that one could do specific computations in groups, and also groupoids. So I began in the 1960s to look for higher dimensional versions of these groupoid methods, again with the aim, or hope,  of higher dimensional nonabelian calculations. Of course we were well aware of all the laws on paths, or singular simplices,  up to homotopies, e.g. Kan extension conditions, but it seemed difficult to get computational  information directly at the path space or singular complex level. 
What was surprising, and took a long time to realise, was that we could do these higher dimensional strict groupoid methods, using certain homotopy classes,  for certain structured spaces, particularly filtered spaces (11 years), and later $n$-cubes of spaces (Loday) (17 years). In the filtered space work, the insights of model theory have also proved very useful - I think they have not yet been used in the $n$-cube situation. Grothendieck was amazed when I told him in 1985 (6?) that $n$-fold groupoids model homotopy $n$-types (Loday's theorem). Since we can use this idea for specific nonabelian colimit  calculations in homotopy theory with the aid of a Higher van Kampen type theorem, I am happy as an old man to rest with the use of strict multiple groupoids of various kinds suitable for the problem at hand. Of course in the proofs, the relations between the weak (spaces of maps) situation and the strict one is crucial.  
I see these ideas as another contribution to the tool kit  of  algebraic topology, and some younger people are using them.  
It seems a useful, but not obligatory,  test of a theory to ask if it can in some cases produce some numbers not previously available. 
February 25, 2015 Here is a link to a talk in Galway, December, 2014, giving further background to this answer, particularly in relation to the history of algebraic topology. 
November 8, 2015: Here is a June, 2015, presentation on 
A philosophy of modelling and computing homotopy types, 
which gives the background to  using,  for specific  computations, some   algebraic models of homotopy types in the form of strict higher groupoids.  It is the strictness which leads to precise colimit computations  in homotopy theory, generalising those well understood for  fundamental groups. One construction these ideas have led to is a nonabelian tensor product of groups, whose current bibliography has 138 items, mainly by group theorists. 
Jan 21, 2017 Parts of the above mentioned presentations have been expanded into  this paper;  it  also discusses  the increasing use of cubical sets with connections, which have been little used so far in theories such as quasi categories, but which in some areas have advantages over simplicial sets.  
A: I find some of this exchange truly depressing.  There is a subject of ``brave 
new algebra''and there are myriads of past and present constructions and calculations that 
depend on having concrete and specific constructions.  People who actually compute 
anything do not use $(\infty,1)$ categories when doing so.  To lay down a 
challenge, they would be of little or no use there.  One can sometimes use 
$(\infty,1)$ categories to construct things not easily constructed otherwise, 
and then one can compute things about them (e.g. work of Behrens and Lawson). But the tools of computation are not $(\infty,1)$ categorical, and often not even 
model categorical.  People should learn some serious computations, do some 
themselves, before totally immersing themselves in the formal theory.  Note that 
$(\infty,1)$ categories are in principle intermediate between the point-set level
and the homotopy category level.  It is easy to translate into $(\infty,1)$
categories from the point-set level, whether from model categories or from
something weaker.  Then one can work in $(\infty,1)$ categories.  But the 
translation back out to the "old-fashioned'' world that some writers seem to 
imagine expendable lands in homotopy categories.  That is fine if that is all 
that one needs, but one often needs a good deal more. One must be eclectic. Just
one old man's view.
