There is only one reason why an individual program would need to be 64-bit instead of 32-bit, and that's so that it can access more than 2 or 3GB of memory. This can be important for programs that hold a lot of in-memory data such as databases, photo or video editors, but for most other programs 64-bit is not needed.
And as long as a program doesn't need that extra memory, then it will run just fine as a 32-bit process on 64-bit Windows. There's no reason not to use a 32-bit program except when it uses too much memory.
My take is that 64-bit IE has been included in Windows 7 so that the makers of plug-ins such as Flash and Silverlight have a platform they can work towards in upgrading their own products to 64-bit. That prepares for a future in which all programs are 64-bit and 32-bit support can be dropped from a future version of Windows a few generations from now, much as 16-bit support was dropped from Windows 7.
But right now, for the general population, there's really no reason to use 64-bit IE.
The OS itself is another story, because it needs to support lots of different programs running at the same time. While few individual programs need more than 2GB of memory, all of them together often do. So 64-bit Operating systems are rapidly becoming the de facto standard.
As far as application programs go, the problem is that until everyone is using a 64-bit OS the software makers are sticking with 32-bit programs so that they can still sell programs to everyone. Producing two versions (32- and 64-bit) is extra work and support, and producing 64-bit-only means you're prevented from selling to a whole swath of users of 32-bit operating systems. There are only a few vendors whose software requires a LOT of memory who are making 64-bit versions available - Photoshop is a good example of this.
I expect that the next desktop version of Windows will be 64-bit only, and a few years after that when the share users with 32-bit operating systems drops to less than 10 or 20% we'll see a wholesale shift to 64-bit applications. Until then, 32-bits works just fine for the vast majority of programs and there's no need to feel shortchanged when you run them.