Why connect arduino with Rasberry pi?

Interfacing hardware is easier with the Arduino: more available I/O pins capable of sourcing and sinking more current, 5 V I/O, analog inputs, very predictable timings, etc. The Raspberry, on the other hand, is a real computer with a real OS, has a lot more computing power and memory, has Internet connectivity, can easily run “standard” web servers like Apache or nginx... or mostly any Linux software for that matter.

Some projects can benefit from using both. E.g. if you want to make a web interface for some low-level hardware sensors.

Why add an Raspberry Pi to an Arduino?

There are some things that simply are beyond the possibilities of Arduino. The first thing that would cross my mind is video processing, HDMI output, decent sound etc.

A Raspberry Pi can be used to add some features to your project aswell. If you make an Arduino controlled robotic arm, you could add an "user interface" to it, using a Raspberry Pi.

Why add an Arduino to an Raspberry Pi?

Some may argue that the Raspberry Pi is much more capable as an Arduino, in every form. Though, the strengths of Arduino are:

• Ease of use. (For GPIO on Raspberry Pi you'll likely need to install something, to make it work with your favorite programming language.)
• Real time / deterministic timing, this is one of the best things of Arduino. Given that an Arduino has no "operating system" that can induce random (tiny) delays in your program (other than the millis()/micros() interrupts).

Why do any of this? You could choose an Arduino OR Raspberry Pi

One thing you might not think about is, that by joining an Arduino and a Raspberry Pi, you're joining two important parts of Embedded Systems. Which you can learn a lot from.

Microcontrollers and Single-board computers. This really has a good educational effect. Since you'll not limit yourself to one device or "way of thinking".

Moreover, the interfacing between two devices is often important and will bring some extra problems. (And solving them gives extra knowledge/experience)

I prefer to use arduino type boards as data acquisition units then transfer the data to an actual computer (or raspberry pi) in most of my projects. Arduinos are, in my experience, much more tolerant of fault and accidental crossed circuits than the R pi. Thus is especially true of the ruggedized or industrial arduino versions.