[Verizon Samsung Galaxy S III] funnyperson1:
That's probably the oddest cross-shop combination I've ever heard of. The iPhone5 and NoteII are both excellent but completely different phones for people with different goals in a phone.

I don't have a NoteII to compare to but my Wife has an iPhone5 and I feel like I have a good read on the differences between the 5 and the SIII.

Screen:

The screens on the iPhone5 and SIII both have their tradeoffs. I love the huge screen on the SIII it allows me to browse without constantly zooming and panning like one needs to do on the iPhone5. This is even more true because most iOS apps haven't been updated and show black bars on top and bottom so like 1/4 of the screen is black in popular apps like whatsapp and engadget.

One the other hand the iPhone is definitely more manageable with one hand and has more accurate color reproduction. If you hold a white page up side by side the GSIII looks yellow in comparison.

Overall I prefer the GSIII screen size, but it's really a matter of personal preference.

Productivity and Capability:

There's simply so much more you can do with Android than you can do with iOS, especially with a rooted and unlocked phone with custom roms. The notifications pulldown, Google Now, widgets, quick settings/toggle, popup browser, etc. all let you do more things faster with Android than you could do with iOS.

While iOS is solid it feels dated with little recent innovation in the actual UI. Almost all of the power of iOS comes from it's app store and third-party support for things like passbook. In the end Apple's homescreen is like a prettier rendition of a Palm Treo launcher from 2003. Siri is cool but imho Google Now is better.

App Selection:

As hinted above, Apple's real strength is in the app store. Overall they have more and better quality apps. There are almost always comparable apps in the Google play store, but they don't always have the same level of polish as the iOS version.

Sometimes though the Google play store has much more powerful apps that would never be allowed in the App store. Things like emulators, apps requiring root, different keyboards, etc. Now that I've used it, I'm not sure how I lived without the Swype keyboard. When I have to type something on my wife's phone I feel like I'm going back in history. I love this. Just today I installed an app that gives you full control of what happens when your proximity sensor triggers.

The restrictions Apple puts on app developers generally result in smoother more reliable apps that don't often crash and can't hijack your battery. On the other hand Google allows much more powerful applications, but rogue applications can hijack your phone and incompatibilities between Android versions cause all sorts of problems. I had to uninstall the Engadget app on JB because it was using 30% of my battery even though it was only in the foreground for 5 minutes.

Performance:

The iPhone will generally give you a smoother, more predictable experience. I think the lack of widgets, app freedom, etc. has a huge impact on this.

While Jelly Bean is butter-smooth for the most part, there are occasional stutters here and there that you don't really see on iOS. A lot of times I will find myself swiping a number or something and the action will take place a second later. Apps themselves seem to perform better on iOS. For example TempleRun stutters and misses swipes after running for a while. This was never an issue on my old iPhone4 and the SIII should be between 2-4 times as powerful.

Battery Life:

In my experience the iPhone5 has more consistent and generally "better" battery life when compared to the SIII. I put better in quotes because in certain use cases I'm sure the SIII would achieve as good or better battery time. Whenever someone has a battery issue they're told to turn off location services, latitudes, gnow, weather widgets, monitor your wakelocks, etc. However with the iPhone5 you generally don't have to worry about that stuff, if you aren't actively using an app it won't use much of your battery. My wife and I have similar usage patterns and she always seem to have more battery left than me at the end of the day.

Overall the iPhone is a solid but boring device with reliable performance and battery life where the SIII can be a bit inconsistent but delivers a level of customization, productivity and power that iOS can't touch. I think power users will prefer the SIII/Note2 and the iPhone is more for people who expect a consistent experience from their phone.

These observations are probably valid for the Note2 as well. The difference in screen size is even more drastic, performance should improve with the NoteII, battery life is likely about the same, and the OS/app selection remain the same.