Well, as I know, Arduino is quite different thing. It's just a PROTOTYPING board, not a computer. You will need to flash your code into it via USB every time you need it to be executed. RPi is a full-functional linux-based computer. You can just connect RPi to the ethernet via USB Wi-Fi module and controll all your stuff via SSH WITHIN RPi wirelessly. RPi is much more convinient and functional then Arduino. 

(it can be that I don't know some Arduino products)

There are new boards based on Arm like Udoo Quad, which provides more than 4 times the processing and memory of Rasberry PI, as well as sata interface, but also integrate arduino on the board. You want both functions. I understand that Wolfram is looking to expand the Arm universe and has a Beta for BeagleBoard, I sure hope they include Udoo.

There are also various boards gertduino and such that allow you to integrate an arduino and a Rasberry PI.

All sorts of scientific analysis and measurement tools are in reach with a Udoo Quad and varius sensors.

-Kimberly Peacock

There is a fundamental difference between the Arduino and the Raspberry PI. The Arduino has a Real Time Operating Systen and the RPi does not. Linux like Windows and MacOS has a stochastic operating system meaning that the sheduler moves different processes in and out of execution according to a events and work load etc at a given time. This is basically a random process so the RPi cannot do anything that requires really exact timing. For operations that work over second to hours this hardly makes a difference but if you try to sample fast waveforms serious and unpredictable errors can result. The Arduino with its real time operating system does not suffer from this problem so is much better for data aquisition. The flip side of this of course is that the Arduino has very limited comutational ability compared with the RPi Linux system where there are thousends of applications of all kinds. The two boards therefore work very well with each other if you try to exploit the advantages of each.

@bjs

It is really an overstatement to say that Arduino, in its' AVR incarnations, has an operating system. Rather, it has statically linked runtime and libraries, which exist explicitly mostly on source level, but not really on machine code level. This doesn't mean Arduino wouldn't be trivially capable of hard real time operation. Partially this is supported by cycle-predictable nature of AVR microcontroller.

Raspberry Pi with Linux - Mathematica on top - is a different beast. Even if you ditched the kernel, you couldn't run similarly cycle-exact real time processes, because CPU and its' interfaces are not designed for this.

For best of both worlds, you probably want to use Arduino as a sort of real time slave under control of Raspberry Pi (or other fully equipped computer with sufficient interface to Arduino). Especially if you want to achieve microsecond-level (or, realistically, anything that has hard demand for below one second) response latencies, splitting tasks between the microcontroller for that and complex computation for the "real" computer is likely to be mandatory. At least if you're a hobbyist.