So which one is better and why?
In terms of convenience and features, the 600EX-RT is better. It has more features, a built-in radio triggering system, and a much better (less cryptic) UI, since it has a dot-matrix LCD display.
I just need to get as much flash power as possible for outdoor and indoor photography.
If you need as much flash power as possible, you should be looking into the Godox bare bulb Wistro flashes or studio strobes and battery packs, not speedlights and their little AAs. Speedlights are the low-end of the totem pole when it comes to flash power/light output.
With the 580EXII and 600EX-RT, however, this is a dead heat. They're exactly as powerful as each other, despite what the guide numbers say. The 600EX-RT can zoom to 200mm, while the 580EXII can only zoom to 105mm. Zooming pushes the flash tube further back into the head to give a tighter, narrower, more focused output. And Canon measures the guide number at the tightest zoom to get the highest number. Which is why the 600EX-RT has a spec. guide number of 60m (it actually varies from 26 to 60 throughout the zoom range), and the 580EX models have a guide number of 58m, but the power output is "virtually the same", according to Syl Arena.
Are they a lot different from each other? I couldn't care less about the 600EX-RT's radio capabilities; I won't be using that feature at all.
If you remove the new wireless RF protocol, then in capability, they aren't hugely different from each other, only in usability. The feature set is roughly the same, but the UI is quite different. For example, with custom settings on a 580EX/580EXII, most of us either have to change them through the camera menus (assuming we have a Digic 4 or 5 camera that can do this) to remember what they are, or carry a little printed card around that tells us what the settings are, because basically they're listed by number, without names, as are the options. Turning on the optical wireless capability requires remember to push the ZOOM button. That sort of thing. The dot-matrix LCD and soft labels for the buttons on the 600EX-RT erases a lot of that confusion.
Syl Arena has the best information on the difference between the 580EXII and 600EX-RT.
See also this Syl Arena video for CPN Europe for a sense of the UI differences.
However, for most of us who have read the Strobist and like doing studio-style lighting with our speedlights, the built-in RF capability (and additional features to Canon's wireless protocol) is the biggest reason to buy the 600EX-RT.