Re: Why buy Raspberry Pi?
The key distinction between the two is that Arduino is a microcontroller platform, where Raspberry Pi is a microcomputer.
As some examples of the difference, microcontrollers tend to have better low level peripheral support, like ADCs and timers, and they also make it easier (relatively speaking) to do real-time programming. They tend to run on less power, and make very low power applications possible if you put in the work to control what hardware is running and when. Microcomputers have OSes and higher level programming and debugging tools, and much more CPU power for calculations, UI interactions and other typically 'computery' things. The Pi's USB host, Ethernet and audio/video output are relatively difficult or expensive to duplicate on an 8-bit microcontroller like Arduino.
I always summarize it roughly like so: a microcontroller is a set of peripherals with a small programmable core gluing them together, where a microcomputer is a computer with a bit of I/O attached. Obviously there's lots of room for overlap in the middle there, so as Franklin suggests, it comes down to your own project idea in the end.
You can certainly use them together too, to get the strengths of both. Usually I'd expect to see that as a Pi running most of the software for an application and using the Arduino as a controller for precisely timing peripherals, e.g. a robot with the Pi performing computer vision and other compute-insensive algorithms and an Arduino driving stepper motors and reading quadrature encoders.