I have searched the web for the answer but I can't seem to find an answer.

Can someone please tell me the difference?

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2 Answers

If the data in the wiki is correct, it appears that Intense Rainbow Dye is simply a brighter (more vibrant) variant of Rainbow Dye (these links show the dye applied to a player's clothing). That said, the visual difference appears to be very minor. Intense rainbow dye can be crafted from two rainbow dye.

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The Intense Rainbow Dye seems to produce a more saturated colour effect than the normal rainbow dye. This means that the intense dye appears brighter, richer and fuller. This is true for all variations of Intense dyes and not just for rainbow dyes.

Below are pictures of me wearing the creeper outfit with rainbow dyes. In the first picture, the head has the vibrant rainbow dye while the chest has the normal dye. The next picture has the two dyes are reversed:

               

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