FEM for fluids historically was limited to flows without dissipation, so that the same matrix solver as used for solid mechanics could be applied. Finite difference techniques, being a time-based solution method, are more readily (understandably?) applied to time-varying problems like fluid flows, thus were the first techniques applied and have the benefit of years of development.
Around the time I was taking courses (back when we still rode our horses to the university, and the snow was 12 ft deep, uphill both ways), there were formulations for "stiffness matrices" in the finite element method that incorporated linearized Navier-stokes equations, and a few years later, newer solution techniques that would allow efficient solving of these more complex equations. A problem with the FE method is that you must specify the element type that corresponds to the flow field you are modelling (i.e. subsonic, trans-sonic or supersonic) because the equations change form from elliptical to parabolic to hyperbolic as the flow transitions thru those regimes. In a finite-difference code, the solver can be set up to determine for itself what solution method is required (but it didn't used to be that way). There are probably some nifty auto-mesh routines developed in recent years that can do the same thing for FE codes. All of this is moving towards (as in solid mechanics modelling) having the ability for users to do "push button" modelling, without any idea in their own mind of what to expect...with the usual caveats about GIGO. If that sounds like I'm advocating that you take some fluid mechanics courses before proceeding with fluid modelling, then yes, I am beating that drum, the same way solid mechanics courses will help you know when not to believe your FE stress results.
As regards k-e turbulence modelling, its use is preferred primarily because it works reasonably well, is reasonably computationally efficient (i.e. does not require modelling indivdual gas molecules), and was one of the first decent models to do so. Thus, its development progressed pretty rapidly because people used it and gained familiarity with it. Some development has been made of a k-w (kappa-omega) method that does not require as drastic a near-wall grid density, but it has its own problems too. Turbulence and boundary layer models have a long way to go before they will be truly user-friendly, much less amenable to "push button" analysis, imho. You really need to read some books, take a course, and spend a summer reading journal articles just to get reasonably familiar with them.
The biggest topic in fluid modelling lately (well, ok, 7 years ago) is that dropping costs of memory and higher processor speeds allows simulation of the "full-blown" full Reynolds-Averaged Navier-Stokes equations...but as of several years ago, the simulations had only progressed thru some limited laminar and laminar-turbulent flows; these problems were being set up and run on massively-parallel supercomputers (Beowulf clusters) and still took weeks to run...but the output was truly beautiful; you could overlay Schlieren images from the 1930's and see 1:1 correspondence from the simulations. One more step of Moore's Law, and look out, we may be able to do some of that on our desktops...or at least start seeing more rapid development of cheaper simulation methods by comparing (say) k-e models to the full RANS simulations, etc.
And, in the end, you still need to compare your results to real-world test data before anybody (including you) will trust them. Same as for FEA.
