Which biome has the greatest biodiversity, and why?

The answer depends on how you define biomes and what you consider to be a biome.

For example, some divide terrestrial biomes up (taiga, grassland, temperate forest, etc) but consider the aquatic biome to be one type of biome, including deep ocean, coral reefs, and intertidal zones all as one (see image below).

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Thus, the aquatic or marine biome would have the greatest diversity in a large part due to the size of this biome. Using this classification, aquatic biomes cover an immense area and therefore a very large number of species.

However, if we look only at terrestrial biomes, tropical rainforests have the greatest biodiversity.

This tremendous diversity is due to a few factors, some of which are still debated. In general, diversity tends to decrease as one moves away from the equator and towards the poles. Tropical rainforests are found right around the equator. Some argue that a uniform climate in regards to temperature is important for biodiversity. The amount of available energy or productivity is also thought to contribute to diversity, and tropical rainforests have a large amount of both. Thus, this energy is available for species to use and diversify.

Primary productivity across the globe with greater numbers shown in yellow and red and lower numbers in purple and dark blue.
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Tropical rainforests are also one of the oldest biomes on the planet. Having existed for a long period of time, species have had more time to diversify.

To read more about why tropical rainforests are so diverse, check out this extensive page.