As far as I know the main difference is that for amis you do it in a spiral, while to make it flat you need to do it so that each row is separate.
Actually, that's not quite true. I've made many flat circles in a spiral. The number of sections (i.e. the number of times you repeat the increases) affects the shape of your piece. If you only do it 4 times, you'll get a sort of pointy egg shape. If you do it 6 times, you'll get the nice round ball that most amis use. If you do it 8 times, you'll get a flat, shallow dish or a flat circle, depending on how tight your gauge is.
I crochet a lot of bowls and vessels, and I play around with the number of sections to get the shape I want. You can see some of my stuff here on Flickr
, to get an idea of the different shapes you can get this way.