Analytic solutions are not necessarily easier to calculate.
Analytic solutions tend to involve special functions for which the computer can only compute an approximation anyway. Have you ever tried to write code to evaluate the error function over the entire domain of floating point numbers? (Yes, I know, it's now in the standard library; ten years ago, it wasn't.) That's one of the easier ones.
Even if there are no special functions, analytic solutions are still often harder to calculate if the problem is big enough. Think of solving systems of linear equations, one of the standard workhorses of numeric programming. We're talking really big ones; hundreds of thousands of equations in hundreds of thousands of unknowns or bigger. In the real world, this problem would almost certainly be solved using successive approximations, even though high school students know how to solve them analytically.
Finally, and most importantly, the problem statement is usually an approximation. Take the OP as an example. What this kid almost certainly solved was an analytic solution to the problem of a particle in a gravitational field with linear air resistance. Well, air resistance is not linear. At low velocities, and for projectiles with a sufficiently small cross-section, it's close enough. But it's still an approximation.
The advantages of analytic solutions are almost always not computational. What they buy you is understanding. The methods of obtaining the solution, and the form of the final equations, often reveal some deep insights about the problem. For many situations, that's far more valuable. And it's certainly something that no computer can give you.