Diversity implies difference. A diverse forest has many different species, both plant and animal. Often the more diverse a forest, the less chance there is that losing a single species will result in the loss of an entire forest. Many other species still survive and life goes on. Forest monocultures, or those with a single species, are more at risk from a new disease or insect destroying the entire forest. If there is only one species and it becomes extinct, you are left with nothing.
Because of time, native forests are usually more diverse than non-native forests. Time favors diversification, as new species evolve to fill specific niches or tasks in a forest. Gradually an interdependence is created among species which co-evolve together. A bird, for example, may evolve to feed on certain insects. These insects may in turn pollinate only certain plants, which in turn create shade favorable to the generation of young tree seedlings. This tree might be the only tree in which the bird can nest, and therefor reproduce. In a natural system, populations of all species ebb and flow dependently. Too many birds, for example, can result in less insects, which can result in less trees to nest in. In nature, there is a constant balancing among species and their numbers. Normally this is slow process which occurs over several generations.
When a new species is introduced, however, imbalances can quickly occur. The new introductions often have no controlling predators. Sometimes they are very competitive, and are able to replace the native species. In the example above, this can be disasterous for those species dependent on the replaced native species. The result is entire ecosystems threatened by just a few introductions. Unfortunately, many of Hawaii's forests are living laboratories of such events, as non-native introductions have quickly replaced once native forests.