Issue replacing 27C256 ROM with 62256 RAM
"If I 'reconnect' the PIC again and read the memory it is still
  intact"


This indicates that the data is being preserved, but does not prove that it is stored in the correct locations and being presented to the target in the right order. There may be a mistake in your PIC code, or some address or data lines are swapped. So long as each data byte is held in a unique location it could read back correctly through the PIC, even if stored in the wrong memory locations or bit positions. The RAM doesn't care where the data is stored in it, but your target does!  

Plug the target connector into your EPROM programmer and read the 62256. Does it contain what you expected?