Clover and dandelions work to each others' benefit, and thus have a mutual relationship in many ways. Clover is a legume, and thus supports bacteria in its root nodes that turn atmospheric nitrogen into plant-food nitrogen. This helps all nearby plants, including dandelions. The dandelion taproot mines into the soil, pulling up minerals and nutrients from deep in the soil and nourishing it when their leaves fall and that chiseling taproot dies and decays. One does not necessarily ever take over in this mutual situation.
Dandelions don't get very tall. Clover varieties that grow tall can quickly shade them out, especially if the clover has become quite dense. Whether through shading or crowding, this means dandelions might not grow. However, if the clover is mowed or cut for hay, any dormant dandelion seeds will spring up. Likewise, if the clover variety is very short or not allowed to grow tall, dandelions might still find a niche in which to grow and get enough sunlight to flower and set seed, or may even crowd out the clover instead.
Well-established lawns that grow in thick and remain trimmed might still have some clover, but dandelions have a hard time eking out an existence under those circumstances. In this scenario, however, it's not the clover that is crowding out the dandelion -- it's the thick, quick-spreading turf.
Dandelions are edible by both humans and forage animals. In fact, dandelion flower heads used to be quite popular in summer salads. Dandelions provide a necessary function (mining for minerals and leaving humus-rich wedges in the soil where their taproot used to be) for free in both the lawn and in pastures. With such useful qualities and a mutually beneficial, soil-building relationship with clover, it is often best not to allow clover to crowd out dandelions.