When should I use "cross" and "pass" on radio telephony?
When related to a significant point, there is no difference between pass and cross. For altitude/FL only pass is used.

The phraselogies used by ATC is described in ICAO Doc 4444 (Air Traffic Management / PANS-ATM), chapter 12, phraseologies. The document explains the meaning for ATC, therefore to be understood and used by pilots.


  12.3.1.2 LEVEL CHANGES, REPORTS AND RATES
  
  
  EXPEDITE CLIMB (or DESCENT) [UNTIL PASSING (level)]
  AFTER PASSING (significant point)
  
  
  12.3.1.4 TRANSFER OF CONTROL AND / OR FREQUENCY CHANGE
  
  
  AT (or OVER) (time or place) [or WHEN] [PASSING/LEAVING/REACHING (level)] CONTACT (unit call sign)
  (frequency)
  
  
  12.3.1.10 ADDITIONAL REPORTS
  
  
  REPORT PASSING (significant point)
  REPORT PASSING (three digits) RADIAL (name of VOR) VOR
  
  
  12.3.2.3 MAINTENANCE OF SPECIFIED LEVELS
  
  
  MAINTAIN (level) UNTIL PASSING (significant point)
  MAINTAIN (level) UNTIL (minutes) AFTER PASSING (significant point)
  
  
  12.3.2.4 SPECIFICATION OF CRUISING LEVELS
  
  
  CROSS (significant point) AT ( or ABOVE, or BELOW)
  CROSS (distance) MILES, (GNSS or DME) [(direction)] OF (name of DME station) OR (distance) [(direction)] OF (significant point) AT (or ABOVE or 
  BELOW) (level)
  
  
  12.3.2.8 SEPARATION INSTRUCTIONS
  
  
  CROSS (significant point) AT (time) [OR LATER (or OR BEFORE)]
  ADVISE IF ABLE TO CROSS (significant point) AT (time or level)
  
  
  12.3.3.1 DEPARTURE INSTRUCTIONS
  
  
  AFTER REACHING (or PASSING) (level or significant point) ( instructions)
  
  
  12.3.4.12 TURN OR CLIMB INSTRUCTIONS AFTER TAKE-OFF
  
  
  AFTER PASSING (level) (instructions)
  
  
  12.4.2.3 PARALLEL APPROACHES
  
  
  YOU HAVE CROSSED THE LOCALIZER (or GBAS/SBAS/MLS FINAL APPROACH COURSE)
  


If another wording is used in some country, the differences must be explained in the country AIP, at section GEN 1.7. For USA (FAA AIP), Doc 4444 differences are found at GEN 1.7-23. There are none related.