Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background.

One question is haunting me.

Why do I need to learn to do complex math operations on paper when most can be done automatically in software like Maple. For instance, as long as I learn the concept and application for how aspects of linear algebra and differential equations work, won't I be able to enter the appropriate info into such a software program and not have to manually do the calculations?

Is the point of math and math classes to learn the big-picture concepts of how to apply mathematical tools or is the point to learn the details to the ground level?

Just to clarify, I'm not trying to offend any mathematicians or to belittle the importance of math. From CS I recognize that knowing the deep details of an algorithm can be useful, but that is equally important to be able to work abstractly. Just trying to get some perspective on how to approach the next few years of study.

Is the point of math and math classes to learn the big-picture concepts of how to apply mathematical tools or is the point to learn the details to the ground level?

Both. One is difficult without the other. How are you going to solve equations that Maple can't solve? How are you going to solve it, exactly or numerically? What's the best way to solve something numerically? How can you simplify the problem to get an approximate answer? How are you going to interpret Maple's output, and any issues you have with its solution? How can you simplify the answer it gives you? What if you are only interested in the problem for a particular set of values/parameters/in a particular range? What happens if a parameter is small? How many solutions are there? Does a solution even exist?

Using a CAS without knowing the background maths behind the problems you're trying to solve is like punching the buttons on a calculator without knowing what numbers are, what the operations mean or what the order of operations might be.

Is the point of math and math classes to learn the big-picture concepts of how to apply mathematical tools or is the point to learn the details to the ground level?

I will second Bennett, the point is both. Consider the analogy that learning mathematics and physics is much like constructing maps. First, you will see maps others have created, how details are crafted, norms, what are usual rules, what are the great maps for certain regions. This is the highview.

However, you must be sure these maps are correct. Therefore, you'll go to the places they give you directions to and check if it matches. This is the ground level. You have to make sure you are following instructions correctly, arriving at the same results, be able to walk yourself through the path.

It's the only way you have a firm, solid, sharp knowledge of anything you study. Learning how to switch between the bird's-eye view and sniffing the ground is part of the apprenticeship of anyone in science.

I will end this answer with a quote from Richard Hamming:

The purpose of computing is insight, not numbers.

Many, many reasons...

1. I was in a number theory program in high school that prohibited calculators. You know how many more things you notice about the theory when you have to find ways to figure out the details yourself? Maybe you have to compute something absurd like $3^{24} (\mod 7)$, but you know what? That's actually really easy once you realize the powers start repeating cyclically. You would never learn this if you let the computer go up 24 powers and spit out the number 1 (which I just calculated in my head).
2. You learn intelligence that you will need everywhere. Try saying "well I usually use a computer to do this" in a job interview. In other words, I'm saying that even employers, often, think so.
3. You will have a hard time solving related problems or even modeling properly with a diffeq if you do not understand how diffeqs work. Say the computer can't just quite solve it; can you reduce it to a problem it can solve? Can you try solving it by hand and see where you get stuck to understand why this is an interesting diffeq?
4. The best way to make sure you understand the high level concepts is by working out a detailed problem of tolerable but significant difficulty. Otherwise you are just cheating yourself that you can understand high level concepts.
5. The details will reveal to you why the high level concepts are important, and you will remember them better. If your theorem has X, Y and Z hypotheses, it may be hard to keep track of this, unless you prove the theorem or solve a problem and explicitly see where you need X, Y and Z hypotheses. Perhaps you can even visualize why you need them, so if a different problem looks different in your head, you will realize you were about to do something illegal.
6. The devil is in the details. When working out the details may be the time you notice you are missing a prerequisite or modeled it wrong. e.g., "At this point in the problem, I would usually move this term here, but that would imply the water is bending upward which isn't happening in my physical problem... did I make a mistake somewhere?"
7. It's interesting. Maybe you won't be that interested in it, but someone else in your class is. Or maybe you will be surprised and find one technique particularly cool, and learn a bit more about that area of diffeq and the physics behind it and for the rest of your life be better at dealing with those kinds of problems. But that person will go on to get a Ph.D. in mathematics, and work for Wolfram and develop Mathematica, so the next generation of professionals will have fewer diffeq's the computer cannot solve.

Without knowing the details of a process, it is extremely difficult to program tools yourself that compute this process. Put more succinctly, without understanding an algorithm, it is nearly impossible to implement the algorithm. This is not nearly its only justification, but I would wager it is the most relevant, given your background.