The RaspberryPi is supposed to be a computer you can use like a regular one. Rex seems to have a dual role: It's designed to control a robot (lots of pins and extension boards) and to act as a general purpose computer. So it sits somewhere between an Arduino and a RasPi.
The BeagleBone Black also sits in that range in that it is both an ARM-based general purpose computer but also has 2 real-time programmable microcontrollers onboard. Which isn't to say that this Rex project has no value, if it is produced it would be a much easier out-of-the-box experience for someone to just plug in some garden variety I2C motors and get going than it would be on the Black, but this sort of combined computer/microcontroller design isn't unique to it.
IO expansion and power handling is one differentiator from the Raspberry Pi. The Raspberry Pi has a limited GPIO breakout (useful for many applications, but not tailored for robotics). And, besides not being designed to drive peripherals, the RPi's power system has some issues[1] that even make hosting USB devices a pain.
It does seem like they could target the Beagle Bone Black and focus on the software, but they probably wanted more control over the power and peripheral connect strategy than the BBB provides.
The processor they chose includes an integrated DSP, which could be useful for sensor data processing--or even motor control. The original BeagleBoard offered an OMAP with a DSP, but TI is discontinuing the OMAP line[2] and the newer BBB uses a Sitara A8 processor without the DSP. I didn't see the Rex's processor choice listed on the kickstarter page, but I'd guess it's something like TI's DM3730[3].
In summary, they seem to want to design a board specifically for robotics rather than building upon an existing general-purpose platform. This will give them more control over the power and peripheral strategy that will allow them to achieve a higher level of integration. This could also simplify software by limiting the number of busses and "tacked on" peripherals.
The founders come from Carnegie Mellon University, which has a first class robotics program. Also check out the CMUcam[4] project--an open source, computer vision module. The Pixy (CMUcam5) was funded[5] in September and looks really cool.
Edit: Check out the Rex page on the Alphalem site[6] too. It says that the "Rex can supply up to 20A directly from a 6-12V Ni-MH battery to connected devices. That's more than enough to power a couple servo drivers, Arduinos, rangefinders, and full servo load on an 18-servo hexapod robot!"