Without turning this into a religious discussion
and also keeping things overly simplified, the basic difference is that Android is an "Open" platform with lots of different companies and individuals involved, while Apple controls the hardware, software, and apps on the iPhone.
So Apple designs the hardware and controls its production to their specs and they write the operating software to match that specific hardware. Apple also tends to hold back features until they feel that they are thoroughly tested and truly ready to be released to the public. So you know that what does get released generally will work without any problems. They also have ultimate control over what third-party apps are released through the iPhone App Store. That's not a perfect scenario as some "bad" apps can sneak through, and some "good" apps are excluded by Apple, but generally, anything you download to your iPhone has been thoroughly tested and vetted by Apple.
Android isn't so much a phone as a "platform". Google writes the main software, and anyone is free to manufacture a phone or tablet based on that software. So you get a wide variety of hardware running Android, and most of the hardware manufacturers also tweak the Android software to fit their own purposes. And then the phone carriers also add their own software and apps on top of that, mostly designed to keep you tied to their own network.
What folks are figuring out now is that even if Google releases a new version of Android, the phone manufacturers and carriers don't necessarily have to offer it on their phones. Since they've generally done a lot of software tweaking, it ends up being a significant amount of time and trouble to port the new software to "older" phones. We've seen cases where Android phones and tablets that were the killer hot sellers less than six months ago are simply unable to upgrade to the latest Android software, meaning that any new features, performance improvements, and bug fixes are simply unavailable. You want the new stuff? Then buy a new phone, even if your two-year contract isn't up yet . . .
This means that no two Android phones are alike, and also that they don't necessarily work the same between carriers, between models, and sometimes, between different apps on the same phone. That can be frustrating when all you want to do is go to the Home screen, and sometimes that's a swipe left, sometimes it's a swipe up, and sometimes it's a specific button located at the top (or maybe the bottom) of the screen that may be labeled Back or Home or something else entirely.
So you may find a killer Android feature or app, but that doesn't mean that it will be available on any other Android phone, or that it will work the way that you expect it to. Or that it will even be available in a future software update. So you get that killer 4.5" screen, but what you got when you bought is simply all you'll ever get, regardless of how technology marches on.
In contrast, Apple may only produce one 3.5" screen size, but basically all software upgrades are still available to every iPhone they've ever produced. Sure, there are sometimes hardware limitations (such as processor speed or limited memory), but generally, iPhones aren't abandoned or orphaned after a few months, or even a few years. Hell, I still have an iPhone 3G (2nd generation from ~2008) that still runs just fine, although I do see speed improvements in my iPhone 4, as well as much better screen sharpness and resolution.
And Apple is very specific about how the carriers interact with the iPhone. Basically, the carriers have to agree to take the iPhone as is, without adding any additional bloatware or crapware onto it. Or else they simply don't get the iPhone. You may call that limiting, but it means that consumers know they'll be getting the exact same well-tested iPhone regardless of who provides their voice minutes and data.
Speaking of spyware, a firm called Carrier IQ has recently been making news after their software was found to be embedded in almost all smartphones on the market today. At the least, this software tracks a phone's performance on the cellular network and sends that data back to the carrier so they can improve their network efficiency and reliability. But what some phone hackers have discovered is that the Carrier IQ software can have full access to everything you do on your phone. Contact lists, call logs, and even individual key presses, meaning that it can track your every email and text message.
Obviously, this caused quite a stir and has huge privacy implications. Once it started making the rounds among the tech savvy, and later into more mainstream media, the response from all involved was quite illuminating. Google said they didn't install this software as part of Android, but you might want to look at the phone manufacturers. The phone guys pointed their fingers at the carriers, and said they had no control over what the carriers installed. The carriers said it wasn't their fault, really, and you should talk to the manufacturers about what data was being collected and sent back. And Carrier IQ said they were only providing the services that the phone guys were asking for, while deftly avoiding the question of who was collecting the data, and exactly what data was being collected. Note that this software was embedded so deeply that it was not obvious to any phone user, it was just there and turned on as soon as the phone was activated, with no chance at all to opt-out or even see what data was being collected and sent back to who knows where.
Apple has also admitted to using the Carrier IQ software. But it is turned off by default, meaning a user has to specifically opt-in to use it. Their statement simply said that they had only used it to track network performance and monitor application crashes, that the data was collected anonymously, that they had already phased most of that out in the last software update (well before this story broke), and that the next major software update would have all traces of Carrier IQ removed from the iPhone. That response seems very telling to me, especially compared to the pointing-fingers-blame-game that the other phone guys are playing.
"Open" also means that anyone who wants to can write and deploy Android apps. That's very freeing since there's no "Nanny" or "Gatekeeper" involved to limit things. But since these apps aren't vetted then anyone can write anything they want, good or bad. There have been several documented cases of viruses and malware posing as free, useful Android apps, such as a Free Wallpaper apps that have access to your complete contacts list and phone logs, and that phone home to some server somewhere without the user's intervention or knowledge.
In contrast, Apple is very open about what data is collected and where it goes. You have to physically "opt-in" to such things, as they are turned off by default. The first time you set up your iPhone, iTunes asks for your permission to collect and analyze Diagnostic and Usage data, which is only sent to Apple and nowhere else (I generally leave this off). It is obvious right up front, and trivially easy to opt-out. Also, the iPhone can track its location, but again, Locations Services are turned off by default. And if you do decide to turn them on (I generally do), you have to give permission individually to each app that wants to use that location data. So if an app that finds free wireless hotspots near you wants to know you current location, that makes sense. But if a free puzzle game wants to know exactly where you are, there's no real reason they need that info so you can just say no when you first open the app and the app is then blocked out of collecting such information (if that app even makes it past the App Store approval process in the first place).
There are many more differences, of course. And there are advantages and disadvantages to both approaches. I find that the phones are taking the same path that personal computers did a few decades ago. Apple makes the hardware and software, so it has tight integration. Maybe it doesn't do every possible thing under the sun, but what it does do just plain works.
Whereas Google is writing the main operating software but doesn't produce any hardware (much like Microsoft). So you have dozens of hardware manufacturers all working to different specs and different priorities, meaning that the user experience differs dramatically between phones and carriers, since there is no truly "pure" Android phone out there. Sure, it's "Open", in both the good and the bad sense of the word.
I also find that the true Android Fanboys tend to be the ones who want to tear into the phone, root things, try custom software and apps, and generally, have the curiosity and the skill to dig deep and play around. That's cool, if that's what you want to do.
But most consumers just want to buy a product and have it work right out of the box. Generally, the iPhone does that consistently, whereas an Android phone can be quite hit or miss.
You can compare it to motorcycle riders. Sure, there are those of us that like to work on bikes, and some of the guys on this site have torn engines out, replaced clutches, rebuilt final drives, and gotten deep into the bowels of the bike. It's something we enjoy doing, and that we like to share with others. Other riders haven't gotten as deep, but have maybe wired in lights and GPS units, changed the oil, and done other simple maintenance tasks. That's cool, too, and there's lots of help and advice available to those who are curious and want to learn.
But the vast majority of riders simply want to get on the bike and ride, without worrying too much about what's going on under all the plastic. No problem then, enjoy the ride. But you'd be better served by finding a product that will work right out of the box, will work pretty much like every other motorcycle out there, and doesn't require a major rebuild just to get it up to speed and make it useful.
Sure, I'm biased here. I've been using Apple products for 20+ years now. But it's because I've done the research, dug into the technology and the software, played with all the different computers, tablets, and phones, and have chosen Apple products because generally speaking, it's a company that I can trust to provide products that just plain work as expected. And, quite simply, that's what I want in a phone or a computer, so it works very well for me.