Running uphill is obviously more challenging than running on a flat surface, but on a hilly course, there is also an equal amount of downhill. If my goal is to have the hardest workout, burn the most calories, etc - does the downhill cancel out the uphill? Do I burn fewer calories running downhill than on a flat surface? Does that cancel out the extra calories I burn running uphill? If so, are there other benefits of running on a hilly course, besides added caloric burn?
No, running downhill does not cancel out running uphill any more than driving in reverse removes miles off the odometer. Will you burn fewer calories expending less effort? Sure. But you are still burning more calories than you would sitting in a chair or walking somewhere.
The biggest thing to watch out for when running downhill is uneven terrain and the added stress on your knees. When you are running uphill, you make contact with the ground faster. That provides more resistance to your quadriceps, which means:
Running downhill in a controlled manner can provide more resistance to your hamstrings--provided you aren't trying to go too fast. However, each and every step is going down a greater distance, which compounds the stress to the knee. In order to get the most work out of the hill, and to protect yourself from injury:
By applying that strategy you will be building some strength on both sides of the leg, protect yourself as much as possible during the run, and get some serious conditioning in. Strength (more muscle mass) will burn more calories while you are resting than pure cardio endurance training.
Additionally often people will think a downhill marathon is easier, but in fact, is often harder than an uphill marathon. Or conversely, both kind of suck.
I used to wonder about that, and not entirely believe it, until I did the Salt Lake City marathon which starts on top of Big Mountain, and drops 5000 some odd feet vertically, a good 3000 of it in the first 8 miles. It really wacks out your quads, as Berin Loritsch in his answer noted you have to catch yourself on every downhill foot plant.
(Did not help that the race starts at 5:30AM, at 55F, and by the time you finish it can be closer to 110F in the city, in July).
Running downhill is the BEST way to strengthen your quad muscles. Often we build muscle imbalances by continually running uphill which does great for glutes, hamstrings, hip flexors. Downhill pounds the quads but strengthens them.
How to downhill run effectively
Based on the work by Arthur Lydiard I recommend short downhill strides coupled with uphill "sprints" to fully maximize your running strength (which does wonders for endurance, coincidentally).
Sample Beginner Workout
Additionally, add a half mile to a mile of a gradual hill to your long run each week to help with aerobic power.
Finally with hills, recommend no more than 2-3 times per week and always try to keep a minimum of one or 2 days between hill sessions. You will be sore, especially in the early going but by focusing on form and using your glutes, hips, hamstrings and quads, you will get stronger quickly.