Throughout, I will refer to actual dips, not bench dips.
What is the real difference between dips and push ups?
First of all, dips are harder than push-ups. Good. That means they're better at making you stronger. That's why the "Average F'ing Program" you link to advises people to switch to dips as soon as possible. It's a strength program, after all. You do a small number of mostly barbell exercises, nearly as heavy as possible for 3 sets of 5 reps, and you increase the weight every workout. People get stronger on these programs, very efficiently and very quickly. Push-ups get you stronger, but not as quickly or efficiently or as much. Dips are plain harder, and therefore more productive. They're only included as an assistance exercise for pressing strength. (And regardless, the overhead press and the bench are responsible for the vast majority of strength increases; push-ups and dips are a way to do something like those two exercises, but not spend a lot of energy on it.)
Under which circumstances should I choose which exercise?
You should choose dips over push-ups when you want a harder exercise that builds strength more effectively. Push-ups require less equipment, which is nice. They're also easier to do a lot of volume with, so that can make them better for non-strength workouts.
What would be an effective method to combine both of my programs?
I'm not a fan of the "X number of Y exercise" gimmick. 100 pushups, 150 dips, 500 sit-ups, 200 squats...all of these "programs" are just a money machine for people to buy the smartphone apps instead of getting on a real strength and conditioning program. They promote a simplistic view of exercise and encourage people into unbalanced workout routines. For instances, pushes should be balanced with pulls.
If you want to stick with these programs, I'd do pull-ups along with push-ups instead. (Then again, I have doubts that most people could get to 25 consecutive full-range-of-motion, no-kip pull-ups in the kind of time frame these programs advertise.) That would probably be better than putting two pushing programs together. Or, you could do pull-ups and (real) dips.
But more importantly, these programs aren't meant to be combined. A hundred push-ups is a fairly specific goal, and the program assumes you aren't doing another one concurrently. Maybe it'll work anyway, but I wouldn't assume it would. I'd expect to get burnt out doing so much work that overlaps.
If you're willing to branch out into other bodyweight strength and conditioning programs, I'd look to Coach Sommer or Ross Enamait. They have real strength, strength-endurance, and conditioning workouts and programs using bodyweight only.