When I teach a course for math majors (an analysis course out of Rudin, say), I have a more or less clear idea of what the students should take away from the course, having been in their shoes some 15 years ago.
But when teaching a calculus class for a mix of physics, engineering, economics, business, pre-med, and miscellaneous students, I lack anything resembling this clarity. On the one hand, I can teach them how to solve computational problems, but the majority of these students (at least at the two universities where I have taught) have had calculus in high school and already know how to do the computations, making this painfully boring for them and for me. On the other hand, I could prove theorems. Experience shows that the vast majority of students are not interested in proofs, and it's hard to make the argument that knowing how to prove the Mean Value Theorem will help them in their future non-mathematical careers.
Of course, students can potentially benefit from learning proofs because through this, they will learn how to think in a new way. However, the same could be said of learning almost any academic subject, so this does not help us with the question at hand: what material to teach, how to teach it, and what to emphasize.
If I could succeed in convincing my students that proofs were interesting and/or fun, I could make the class a worthwhile experience for them. But I find that this is a losing battle for the vast majority of students, even bright ones. There will usually be one or two students in a class of 50 who become (or were already) interested in the theoretical side of things, but catering the class to the interests of such a small minority seems like a bad choice. Again, with a class for math majors, I don't have to worry as much about the students' interests (though it's always good to keep in mind) because I know what they need to know, and why. I have no such "moral high ground" with the proof of the Mean Value Theorem.
The standard answer is to ride the fence between the two extremes (computations vs. proofs) by focusing on concepts while mostly avoiding rigorous proofs. I have found it very difficult to adopt this attitude in practice, falling most of the time into one extreme or the other. The one useful concept that I instill in all my students is: stop and think about a problem before trying to compute the answer. I accomplish this by showing them cases where the standard methods fail, and giving them such problems on exams. Most students absorb this lesson, and that's fine. But other than that, I feel I have nothing useful to tell these students, and judging by student feedback on evaluations, they feel the same way. (Which my colleagues all tell me not to worry about, because evaluations are typically mediocre at best for freshman calculus courses. But couldn't this be indicative of a general problem?)
To try to make this question more focused: a good answer would give a big-picture idea of what benefits non-math majors (in particular, those who have already learned the computational methods in high school) can take away from calculus, and/or specific examples of concepts that are "just right" for emphasis in a calculus class, and good ways to teach these concepts.
There is a related question here: How to make Calculus II seem motivated, interesting, and useful? But I am more interested in what should actually be taught, and with what emphasis, than in motivating the material per se.
After posting, I noticed the closely related question The purpose of mathematics in a liberal education when it is not a prerequisite to other subjects. My question is more practical and specific: what concepts, or what type of concepts, can and should non-math majors take from calculus?