Good question Nick, I will try to give you my interpretation of this.
When I bench I don't remember ever feeling it in my lats either, I have to do an exercise that incorporates my lats in order to feel them being worked, and the bench does not incorporate the lats. The bench should work your chest, shoulders and triceps and I certainly feel those muscles being worked, but again not the lats.
Then why do you feel it when you are doing push-ups? When doing push-ups your body is being suspended in the air, while when you are doing the bench your back is laying flat. While suspended in the air with your feet holding you up as well, you are using more muscles in order to do so and the back muscles are some of them. In fact many people when doing push-ups when they get tired let their mid section drift downwards thus putting more pressure on their back muscles and like any exercise once the technique has been lost then the exercise should be abandoned.
I hope that this explains it a little better. And if any others want to add or correct anything then feel free to.
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