The major differences between HyperFlash and HyperRAM are:
1. RWDS signal usage
The HyperFlash uses it only as a read data strobe (output from Flash), while the HyperRAM uses it also as a refresh latency indicator and a write data mask (bidirectional).
2. Address Space (CA Bit 46, Table3.2 in the HyperBus specification, http://www.cypress.com/file/213356/download)
The HyperFlash only supports memory space, while the HyperRAM supports both memory and register spaces.
The HyperBus host controllers should take care for above to support both HyperFlash and HyperRAM. According to the K28F datasheet, the data strobe (DQS) is defined as input signal (not bidirectional) so it only supports the HyperFlash.