This kind of thread will be a magnet for a flame war.. Most who have a preference have no technical reason for liking one over the other. In the grand scheme they are very similar in the end effect (and compared on performance).
However, there are differences primarily with the unified shaders (ati uses a cluster of 5 stream processors that can complete 5 or 6 instructions per clock in the best case, and only 1 in the worst).
Today, however I am thinking about comparing mid range and top end cards.
Mid and high end are identical technically as they use the same shader structure.
I'm not sure what you hope to accomplish.. There are hundreds of metrics to compare cards with price, efficiency, and performance on the top of most folks list. Mind you price, while being most important to a consumer is not really a technical metric. One could sell things at a 500% loss making them utter failures technically (at least as far as makable goes) but still the best selling cards in the world.
I originally wrote more, but hate when people simply copy stuff from the internet and don't do their own research, so erased that.
Architecture is very different (how they organize their components, and how it does the math), and their approach to implementation as a card is very different (small chip with modular combinations in both directions vs bigger chip with less options), this influences their production, launch as well as planning throughout the life cycle.
Check launch reviews and then articles about production, that should give you more than enough to figure it out for an essay.
And most importantly they are both similar, just different, neither is objectively 'better', especially since the theoretically better approach to something (be it architecture or production/implementation) may not turn out to be so due to other concerns (and that applies at so many levels of this it's ridiculous).
An essay on how they different yet the same focusing just on the reason they end up the same would be long enough on it's own (research paper length).
technically and real world testing ATI has the high ground, but they have problematic drivers, Nvidia technically trying to get the high ground, very stable drivers, tests show that the newest versions of both are basically the same, one stong here, and weak there, the other vice versa, so it's what you want, bang for the buck ATI is it, but they have "issues" and Nvidia is seriously overpriced