My personal experience, I have never been able to do anything with a heavy oxidized paint job but with the variable speed buffer.
I have the sponge type buffer too, I use it on new cars which haven't been neglected or Goldie with a new paint job.
The rotary type buffer is good if you know what you are doing. No doubt about it. It' quicker and more more aggressive. But if you've never handled one, you can do a lot of damage with it. I've seen it happen. I have done some cars (and a boat!) with a rotary and I've had really good results... But I was lucky that I was taught by someone that was a real artist with one.
The dual action buffer (what you are calling a sponge buffer) is a pretty handy piece. It's not as aggressive, so it will take a little longer, but the proper foam pad (more aggressive) and a good compound will make all the difference. I've brought back a couple really bad finishes with mine. My rotary stays in the cabinet.
IMHO, someone that has never done anything like this is much better starting off with the DA buffer... It will take longer, but the results will be better, it will be harder to screw up the paint, and you'll have a tool that is more versatile when you're done.
Here's a video showing a good method. A couple things to note. One is that he used a clay bar first to clean the paint and the other is that he used an aggressive foam pad. He's also hawking all his company's product, but that's to be expected.
This is a real good "basics" video on polishing paint. This guy is pretty good.
This is an obligatory "clouds in the hood" pic of a black Grand Prix GTP I used to own. The paint was covered in swirls and buffer marks from some "pro" that detailed the car. It was a mess. A few hours with my DA buffer and some good compounds gave me this.
God, I love black cars...