As I understand it, I think the main difference is that when doing body weight excercises, there are a lot of other muscles engaging in the action. For instance, with pushups, you activate the core to hold your form and then use several different muscle groups to perform the actual push up. All in one go. You don't get that when training one muscle group on a machine.
I think both are important though.
In my personal experience, I am also trying to work towards doing a lot of pushups. I've been working on it since October, and at that time I could do maybe 3 modified pushups. Now, I can do well over 50, and 40 negative pushups. I can do a few proper pushups, but my form is not solid, so my trainer has me lifting a lot of weights and doing a lot of core excercises. She also has me doing a 5 min push up challenge 3x a week. 10 seconds - pump off as many as I can, take a 20 second break and repeat. Since I've become strong enough to do these, my progress has acelerated greatly! Oh, and she also has me lower myself into the 90 degree arm position and hold it to engage those particular muscles for as long as possible, then try to push back up.
Rough start due to major carb WD.
MWF: 1 hour run/walk, 1.5 hours in the gym - upper/lower and core
Sat/Sun=Yard/house work, chasing kids, playing
Family walk every night instead of everyone vegging in front of the TV
Personal trainer to build muscle mass & to help meet goals