Level 5 - Level 21 are all the same difficulty, which is the maximum difficulty of the game.
Earlier levels increase in difficulty every 33 seconds until they reach the Level 5 difficulty.
Level 1 starts at Level 1 difficulty (barrels don't steer often, fireballs are slow, etc.)...after 33 seconds it reaches Level 2 difficulty (barrels steer slightly more often, fireballs a little faster, etc.)...so on and so on.
So each level progresses like this, every 33 seconds.
Level 1: 1 > 2 > 3 > 4 > [5] (there is not enough time on Level 1 boards for it to reach Level 5 difficulty)
Level 2: 2 > 3 > 4 > [5] (Level 5 difficulty would only be reached at the last possible moment of Level 2 boards)
Level 3: 3 > 4 > 5
Level 4: 4 > 5
Level 5-21: 5
Difficulty usually refers to three things: type of wild barrels, how easy it is to steer barrels, and speed of fireballs.
Wild barrels: For Level 1 and 2 difficulty, the wild barrel speed and direction is basically random. For Level 3 and 4 difficulty, they are faster and target Jumpman. For Level 5, they try to bounce halfway between their release point and Jumpman's horizontal position.
Steering: Level 1 difficulty steering is 25%. Level 2 and 3 is 50%. Level 4+ is 75%.
Fireballs: Speed increases gradually for each level of difficulty.
Springs are not directly related to the above changes, but are obviously more difficult as you progress. Level 2 is easiest, Level 3 is a little harder, and Level 4+ are the most difficult.