The beaglebone Black processor includes two independent Programmable Real Time Units (PRUs). Hobbyists and professionals are excited about possible use of these units for real-time applications, which is understood. However, if you can have a RTOS (whether for the beaglebone or the raspberry pi), why would you need the PRUs?
EDIT- For information, the BBB has an ARM Cortex A8 running at 1 GHz, with 1.9 DMIPS / MHz. The PRUs are simple RISCs running at 200 MHz.
Linux, even with the real-time scheduler is unsuited to many critical hard real-time tasks with response requirements at the microsecond level, on the other hand it provides or enables a great deal of functionality in terms of UI, connectivity and filesystem support. These things are either not available in an RTOS or are provided at significant cost in high end RTOS, and with much more limited hardware support.
So if you have a system that has hard-real time constraints, but needs more general purpose computing features such a networking, filesystem connection to commercial-off-the-shelf (COTS) peripherals etc., then the PRU provides a solution to that.
On the other hand I can't help but think that this is a marketing exercise on the part of TI to sell more chips. A similar solution has always been possible (and indeed common) using one or more processors to perform time critical tasks, possibly running an RTOS, while UI and connectivity are handled by a single processor with the necessary hardware and memory resources but without the real-time constraints. The PRU device does have two 32 bit cores, but XMOS xCORE devices have as many as 16 cores - with 16 communicating cores, you may not even need an RTOS.
To answer the question...
... directly; you probably wouldn't need them in that case, but you would loose Linux - and your application may need that. It is just one of many solutions to real-time applications using Linux. You pays your money, and takes your choice.