The iPad is fast and snappy because it doesn't have a "file system", like OSX or Windows. Therefore the iPad OS is not stressing with reading and writing information. It is also using flash storage, compared to a hard disk drive. You put those two together and you get a really fast and snappy device.
The A4 processors most significant asset is it's low power consumption, battery life improvements. It is also a "real" processor, and thus can do the job better than the iPhone.
Such a huge misconception. All OS have filesystems including iOS, only difference is that in iOS, the filesystems are sandboxed from the users, meaning they can't access it directly. This means almost excellent security since nothing can interfere with each other and almost hard to infect as well.
What iOS is missing is an application to actually read the file system or an application to create an accessible file system. Apps like GoodReader allows you to access the file system but locked to Goodreader's sandboxed folder only on the iPad.
If you jailbreak iPad, you can use iFile app to access the whole file system with root access.
You're right about the flash, that alone can provide huge performance gains from the typical computer with spinning drives. A4 itself is very fast with low latency access to the RAM (256MB) due to the RAM being packaged directly on top of the A4 chip.
But most of all, iOS is specifically tuned to the hardware. Apple knows every single detail inside the hardware and can optimize the software to fit the spec and remove anything that's not being used. Not to mention, you can't really multitask on iOS (third party restrictions), a full view app has nearly full resource access so it's fast and sleek with nothing else interfering with it.
Imagine you have two computers, both running XP OS, all same parts.
Now imagine PC 1 have SSD in it, no antivirus/firewall/malware crap running in the background. Now imagine you can only run one application at same time.
Compare that to PC2 with 7200rpm drive, full antivirus/firewall/antimalware running with multiple applications at same time.
Take a guess which one will be the fastest, now imagine PC1 without the SSD. Is it still faster? So you can see how the "integration" of both software/hardware can affect the user's experience.
That's what Apple is excellent at.
Also: A4 isn't more "real" processor than the processor in the previous iPhones. They are basically the same technology with some tweaking and latest fab technologies. A4 is faster, customized and tuned for power efficiency.