The statement "you can obtain solution to any DE with any degree of precision" is practically not true for many systems. Despite massive increases in computational power, I don't know anyone who (e.g.) believes that turbulence will be solved any time in the next few decades solely by throwing more computing power at the problem. Through a combination of experimentation, analysis, and computation, however, we have made a lot of progress.
One major advantage of analytical methods is that they more easily allow you to see larger scale trends in the problem. If a computation is expensive, calculating sensitivities can be difficult, and in my experience sensitivities are found in CFD less often than they should be. Computation in some sense is more like experimentation: it gives you a local value. You need to do multiple runs to look at trends.
I think most people also underestimate how far analytical methods can get. In my PhD, I produced an accurate approximate solution to a nonlinear series of ODEs relevant to the problem I was studying. This gave a lot of insight into the problem that researchers using numerical methods for decades apparently couldn't figure out.
By the way, I approximated the unsolvable nonlinear system of ODEs there as a different nonlinear system of ODEs, albeit an exactly solvable system. Most people linearize, but that would have lead to a far less accurate solution in this case. Many nonlinear differential equations can be solved exactly; you can see a catalog of these at this website and in the associated books: http://eqworld.ipmnet.ru/
It's really quite similar to integral tables. You look up the form of the equation you have and they tell you the solution (often with a proof or citation for the proof) or at the very least give you a head start if they don't have the solution.
I'm working on another theory project right now and I intend to use the same "approximate an unsolveable nonlinear system with a solveable nonlinear system" trick here.