Bluetooth: Why can't I pair my iPhone or iPad with another iOS device or with my computer?
Bluetooth supports many different "Profiles", or methods of connecting BlueTooth compatible devices. Since the iPhone first came out Apple has chosen to primarily support hands-free profiles, such as "ear roaches", stereo headsets, automobile integration for sound systems and phone calls, remote control, etc. The specific profiles supported are described here: iOS: Supported Bluetooth profiles. Conspicuously absent from the supported profiles is device to device file transfer. That is, you cannot pair an iOS device with another iOS device or with a computer for file transfer. That's the simple answer. As there are many other ways to transfer files to and from iOS devices (USB cable, WiFi, iCloud, email, MMS messaging, iMessage, plus many 3rd party solutions such as Dropbox, Box.com, Google Drive, etc.) this should not limit the use of your iOS device in any way.
However, with the advent of iOS 8, Yosemite, and Apple's "Continuity" features such as Handoff, SMS sync, and Instant Hotspot (not to forget AirDrop), all of which use BlueTooth, the lack of the ability to pair between devices has created confusion. The answer here is there is more to BlueTooth than "permanent" connections via pairing. The BlueTooth 4 standard has added a connectionless mode (called variously BlueTooth Smart, BlueTooth Low Energy, BlueTooth LE, or just BlueTooth 4) that allows information to pass between applications on devices, but does not require the devices themselves to be paired. This feature was originally used for connecting sensors to your device. Thus, your Fitbit, your Polar heart monitor and many other biometric devices can transfer data to your computer, iPhone or iPad without having to be paired. The New York Times for 30 October 2014 has an article on it: http://www.nytimes.com/2014/10/30/technology/personaltech/bluetooth-improvements -appear-in-more-devices.html
Apple's Continuity features also use BlueTooth 4. This is why older computers and iPhones won't work with some of the continuity features; BlueTooth 4 requires newer hardware. But its also why you don't have to pair your phone and your computer to use these features. For help with using Continuity features (and troubleshooting them) Apple has published this guide: Use Continuity to connect your Mac, iPhone, iPad, iPod touch, and Apple Watch - Apple Support. And for help with AirDrop see: Mac Basics: AirDrop lets you send files from your Mac to nearby Macs and iOS devices - Apple Support