Is there any real difference between using DB for different bench presses and using a bar?

yes and no
Sure they dont use the same equiptment, but they do work the same big muscle group which is chest. However, you should always "trick" your muscles so that it never hits a plateu or gets use to the same routine. Slight changes can easily "trick" the muscle, but you are asking specifically about these two excercises. So dumbells will work the chest but also work your stabilizing muscles, and bar will obviously not work the stabilizing muscle as much. My advice is to switch off between the two, so that your muscles never get use to the excercises. Also be sure to move up in weight if it starts getting easier.

How about floor exercises like pushups? Does it matter if the muscles get used to those?

Straight bar bench press will put more torque on the rotator cuff. Long term heavy bench pressers have a very high rate of RC problems. Dumbells allow the natural movement of the joint. Because of the natural movement and the adduction at the top end of a dumbell press you can put more maximal contraction into the pecs. I use the DB press as the core movement but I do throw in straight bar presses for a change as the above post mentioned.