Here's some data on level growth and what it actually does, as well as how difficulty affects the game. This data is currently incomplete since it's rather tedious to test every boss over and over.
Summary:
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Difficulty does not change weapon damage and does not interact with the character properties in any way.
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Difficulty does not appear to change drops or the number of points gained from collectables.
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The starting health range between Kids-VH is only due to a difference in starting rank. Modifying game memory to start with a different rank results in greater initial health. Gaining rank is enough to close the difference in health between difficulties.
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Bosses have different health values between difficulties.
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Bosses have several "stacks" of health with 64 health each. On Kids diff., most bosses have 4 stacks. On VH, most have 8 (but I have not yet confirmed this about all bosses.)