@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.

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.

I think we won't have to choose one or the other. RPi is good at some things and clumsy at others. Arduino the same. There's enough interest in this, though, that there should be a wealth of hybrid solutions. This is a pretty good technical reference, although this particular page probably supplies way more information than you need.

Hth,
Fred

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)