It depends on individual body mechanics, but generally speaking they both hit a lot of the same muscle groups. I think something more important to keep in mind is that dips are simply harder than push ups, so, generally, you will be building more strength in your triceps, chest, and shoulders from dips than from standard push ups. But push ups definitely have their place; of course, they are the basis for the one arm push up and the handstand push up. Also, remember that a push up is basically a dynamic plank so generally speaking push ups will have more of an impact on your core than dips (although you should still do a dedicated core exercise regardless…unless you’re talking about an advanced progression that requires more core strength like one arm push ups or Lalanne push ups).
In my experience, doing standard push ups in the same workout as dips or handstand push ups will tire out the dip or handstand push up. However, I find that doing dips and handstand push ups in the same workout (with other exercises at least) will not tire me out as much. This seems odd, since they are both harder than push ups. However, notice that dips and handstand push ups are vertical presses at the opposite end of the spectrum, while push ups are a horizontal press. Push ups are essentially halfway between a dip and handstand push up, with less resistance. This means that the push up will more closely approximate a type of load on the muscles that is comparable to dips or handstand push ups (I find this mostly true in the shoulders…front delts, if memory serves) than either one of them will approximate the other.
I don’t think that was quite the answer you were expecting, but I hope it helps!