Why are numerical solutions preferred to analytical solutions?

• Some equations have no finitely expressible analytic solution ($x^5+x+1=0$, for example).
• Symbolic algebraic manipulation is computationally expensive, even when it can produce a usable solution.
• For some functions, even taking the derivative analytically is too difficult.
• You don't always need an exact solution: sometimes you just want bounds on the answer.

Well, I would agree that an analytical solution generally is preferable, if one can find it. The problem is that in many, many problems, finding analytical solutions is very difficult or even impossible. These problems can still often be solved by numerical methods.

Numerical solutions are quick and dirty:

Quick

This means faster than the analytical way, possible in human life time or even just possible.

Dirty

This means not exact like analytical ones and often enough not too wrong.

If you have a system which can correct things during the flight, you can even fly things to comets far away.

Another reason:

Often, our input data has errors, so exact computation is somehow not as good anymore.