Why separate autopilot controls for course and heading?
First, I'll try for the 50 words or less: Course is the line across the ground that you want to go. Heading is where you have to point the aircraft to fly that line.

That's less than 50 words, but I'm not sure it's any better.

Let's look at the obvious first. Let's say, over a short distance, you want to fly a course of 270 degrees true. If you fly the airplane on a heading of 270 degrees, you will fly that course ONLY if there is no wind, which for all practical purposes is never the case. If there was a wind from the north, you would have to maintain a heading greater than 270 degrees to make good a course of 270 degrees.

Actually, you can never say that the course between point A and point B is a single value in degrees unless those two points are very close together. If you draw a line between point A and point B and those two points are not close together, and then measure the course leaving point A (the angle between north and the line as it leaves point A) you will find that it is different from the course as you arrive at point B.

You can, however, tell the autopilot to fly a direct course from point A to point B and it will do it, changing the heading as necessary to account for the wind and fly that course.

So, why does an autopilot need to have a "course" function and a "heading" function. In my experience, the greatest single use of the heading function is when a controller is vectoring you and tells you to fly heading so and so. The second is when a reroute from your existing course is necessary, and you use the heading function to keep the airplane going straight ahead for a bit while you redo the revised course setting you want to use.

If you forget to again use the course function after putting in the revised course, the airplane will continue on the heading it was on while you were updating the course, and you will shortly be off course. That mistake has been responsible for aircraft being hundreds of miles off course.

During the first Gulf War, I heard a Canadian controller inform a USAF aircraft that they were over 300 miles off course when Canadian radar picked them up prior to landfall coming in from Europe.

A captain at the first 747 carrier I flew for got over 50 miles off course and was demoted to First Officer. He had forgotten to switch from the heading function to the course function.

Finally, as an added problem, if you choose to define your course in degrees magnetic, you must also account for the change in magnetic variation as you proceed along that course.

Insofar as item 4 is concerned, you (or the autopilot) is going to change heading as necessary to keep the airplane proceeding along the course as defined by the runway centerline. There's a bit of confusion of terms here in that runway directions are typically referred to as the runway heading rather than the runway course.