Push-ups and dips are some of the oldest pressing movements. They are both body-weight exercises although additional weight can be added for extra resistance. Both movements have the same primal movers - chest, triceps and front delts. Many people are arguing what's better - push-up or dips? Well, it depends.
In my opinion push-ups are a much more natural movement than dips. When you perform dips your shoulder joint is exposed to a great stress. Some people will experience shoulder pain from dips even if they use perfect form and have strong shoulders. It's just the nature of the exercise. I remember the first time I tried dips at home. I used two chairs and did 6-7 reps. I felt like my chest was going to explode. However later I had some training injuries to my elbow and shoulders (not from dips) and I wasn't able to perform dips due to the extreme shoulder position. On the other hand I never had shoulder pain from push-ups. If you think of it you will never press anything in life the same way you do a dip. At the same time most of the pressing movements in sports and so on are done like a push-up. The shoulder joint is in a much safer and stronger position. So in my eyes the push-up is the safer exercise.
Many people will say that the resistance offered by a push-up is too light and it's somewhat weird to perform weighted push-ups. Yes, it's more convenient to perform weighted dips than weighted push-ups but it does not mean that you can't add weight when doing a push-up. I order to add resistance to your push-ups you can just wear a backpack full of books or just ask a friend to place a plate or two on your back. Another great way to do weighted push-ups is presented in the video below.
Of course all of this does not mean that dips are not a good exercise. They are just harder on the shoulder joint. However I believe that they are a better chest exercise than push-ups due to the massive stretch at the bottom. If you don't experience pain when doing dips keep them in your program. At the same time don't think you are doing less work if you had to switch to weighted push-ups due to injuries and so on. Remember that your muscles don't know the difference between push-ups and dips but your shoulder joints may feel it.