In a flight plan, does it make any difference to write an increase of the flight level instead of writing a cruise climb?
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.