In a flight plan, does it make any difference to write an increase of the flight level instead of writing a cruise climb? According to ICAO 4444, a change of speed or level should be written in item 15 of the flight plan according to these rules:


  Change of speed or level (maximum 21 characters)
  The point at which a change of speed (5% TAS or 0.01 Mach or more) or
  a change of level is planned, expressed exactly as in (2) above,
  followed by an oblique stroke and both the cruising speed and the
  cruising level, expressed exactly as in (a) and (b) above, without a
  space between them, even when only one of these quantities will be
  changed.
  
  Examples: 
  
  LN/N0284A045
  MAY/N0305Fl80
  HADDY/N0420F330
  4602N07805W/N0500F350
  46N078W/M082F330
  DUB180040/N0350M0840
  
  Cruise climb (maximum 28 characters)
  The letter C followed by an oblique stroke; THEN the point at
  which cruise climb is planned to start, expressed exactly as in
  (2) above, followed by an oblique stroke; THEN the speed to
  be maintained during cruise climb, expressed exactly as in (a)
  above, followed by the two levels defining the layer to be
  occupied during cruise climb, each level expressed exactly as
  in (b) above, or the level above which cruise climb is planned
  followed by the letters PLUS, without a space between them.
  
  Examples: 
  
  C/48N050W/M082F290F350
  C/48N050W/M082F290PLUS
  C/52N050W/M220F580F620  


Does it make any difference to write an increase in the flight level instead of writing a cruise climb? Are they different flight maneuvers?
They are different things. A cruise climb allows you to continually climb at your own rate to achieve maximum efficiency. This is what Concorde used to do. However this requires ATC to "give you" all of the levels from the bottom to the top of the climb, which is impractical with the volume of traffic.

Instead, we use set levels. We can "step climb" to another level when we become light enough to do it, and this way ATC only has to block off one level at a time for us.