How is the phraseology of IFR clearances different between the US and Europe?
It's not only the IFR clearance, in Europe (where ICAO regulations are used everywhere) the phraseology is very different to the one in the US.
In Europe the phraseology is clearer and easier to understand, in the US it's shorter, less formal but easy to misunderstand.

Some examples are: 


  118.350 
  FAA: one one eight point three five 
  ICAO: one one eight decimal three five zero
  
  Holding over DKB:
  FAA: Hold over DKB to the west.
  ICAO: hold over DKB as published, FL90


To answer your specific question about the IFR clearance:
In America an IFR clearance would sound like this:


  American 590, Ft. Lauderdale clearance delivery, cleared to Atlanta Intl' Airport, ARKES1 departure, then as filed, climb via the SID, expect FL320 one­zero minutes after departure. Departure frequency 126.05, squawk 5523.


According to ICAO (specifically german phraseology):


  American 590, Frankfurt Delivery, startup approved, cleared to Atlanta aerodrome via DKB8S departure route, flightplanned route, climb altitude 4000ft, squawk 1000


So to sum it up, what's different are the parts "flightplanned route", "climb via the SID", and the departure frequency. If it comes to vectored departures it gets even more different, as in the US the clearance just states "via radar vectors" whereas in Europe the whole vectored departure procedure is inside the clearance ("... via vectored departure, after departure runway 18 climb altitude 4000ft on runway track, passing 3300ft proceed direct DKB, flightplanned route, ...").