What is the difference between a functional check and an operational check?
From the ATA spec 100 which is the standard for aircraft maintenance documentation, they provide the following definitions:  


  (c) Following are definitions of three categories of test that shall
  apply:  
  
  1 Operational Test  
  
  a.  That procedure required to ascertain only that a system or unit is
  operable. These tests should require no special equipment or
  facilities other than that installed on the aircraft and should be
  comparable to the tests performed by the flight crews. It is not
  intended that the operational test of the unit shall meet the
  specifications and tolerances ordinarily established for overhaul, or
  major maintenance periods.  
  
  2 Functional Test  
  
  a.  That procedure required to ascertain that a system or unit is
  functioning in all aspects in accordance with minimum acceptable
  system or unit design specifications. These tests may require
  supplemental ground support equipment and should be more specific and
  detailed than an operational test. It should contain all necessary
  information to perform proficiency tests to maintain system or unit
  reliability at an acceptable level, without reference to additional
  documents.  
  
  3 System Test  
  
  a.  That procedure containing all adjustment specifications and
  tolerances required to maintain system and/or unit performance at
  maximum efficiency and design specifications. It shall be
  self-contained and may duplicate other tests. It is normally used at
  major maintenance periods.


I would expect that these definitions are pretty universal in aviation.