Bench presses isolate the chest muscles more (but also work the anterior delt and triceps). Push-ups are a more functional, total body move. They do focus on the chest, front delt and triceps, but your core will be more engaged, and your shoulder girdle involved more.
By "not feeling anything in your chest" AFTER dong the workout, do you mean tha you don't have DOMS (which kicks in 24-48 hours later?). DOMS is not necessary to have a good workout. In fact, several factors that improve recovery (stretching, nutrition, etc.) may reduce DOMS. Tony includes a lot of stretches in his workouts, which may enhance recovery, IMO, and decrease DOMS.
Were the push-ups challenging to do? If so, then you will benefit from them.
Don't base your judgement on whether a workout is effective on whether or not you have DOMS.