Why are Navier-Stokes equations needed? Can't we picture air or water molecules individually? Then, why are Navier-Stokes equations needed, after all? Can't we just aggregate individual ones? Or is it computationally difficult, or inefficient to generate the picture of whole flow?
 A: As apparent from the other answers, the continuum approach of the Navier-Stokes equations makes life easier, by reducing the degrees of freedom. For example, Poisseuile flow (laminar flow between flat plates or in a pipe), is easily solved using the Navier-Stokes equations, but I have no clue how to solve it using just molecular behavior (on the back of an envelope that is).
If you calculate the Knudsen number of a specific problem, which is just the molecular mean free path divided by some macroscopic length scale, you can see if the continuum approach is a valid for the considered system.
A: Consider a standard volume of $1\textrm{ m}^3$ of air. This contains on the order of $10^{25}$ molecules of O2 and N2. 
If you needed to simulate or explain the physics occurring in that volume of air, would you want to model $10^{25}$ molecules and all the interactions between them or, say, 100x100x100 cells based on the Navier-Stokes equations?
Theoretically, it is possible to simulate every fluid flow ever by tracking every single molecule. But direct simulation of turbulence using the Eulerian Navier-Stokes equations requires $Re^{9/4}$ grid points and is thus totally impractical for Reynolds numbers larger than a few thousand. So simulating something with $10^{25}$ things to track is completely impossible.
A: Apart from the experimental and numerical difficulties to track each particle in a macroscopic piece of matter, there are physical reasons.
Theoretically, we can show that in the hydrodynamic regime the exact equations of motion for the collection of atoms/molecules reduce to the Navier-Stokes equations. This is shown in statistical mechanics.
Due to complex molecular effects, including molecular chaos, most degrees of freedom self-cancel and only some few dynamical modes survive. Those modes are collective --i.e. they describe the fluid as a whole-- and robust --i.e., they survive in macroscopic scales of space and time--.
That is, even if you were to trace the individual motion of each particle and next average to describe the collective motion of the whole fluid, you would see that your description is practically indistinguishable [*] from that given by solving the Navier-Stokes equations.
[*] The differences are of the order of the inverse of the size of the system and vanish for a macroscopic system.
