What is the difference between MIL-H-5606 and MIL-PRF-5606 hydraulic fluid?
About 25 years ago, there was a major effort to revise MIL standards and specifications.  The biggest change in MIL specs was to move away from a design or manufacture spec to a performance spec. The downside of the old specs were that they didn't allow for an improved product without getting the spec revised.  By switching to performance spec (what it must do as opposed to how to make it), improvements can be made without spec revisions and contract revisions (that call out the specs) ultimately saving everyone time and money while delivering products that use up to date technology.  

In your case, MIL-H-5606 is the 'old' spec.  The H is taken from the primary application, in this case "Hydraulics".  The current spec is MIL-PRF-5606.  PRF is used to indicate a 'performance' specification.  

'MIL-PRF' and other abbreviations are defined within DoD 4120.24-M, [2], Defense Standardization Program (DSP) Policies and Procedures, March 2000, OUSD (Acquisition, Technology and Logistics).