There really is no difference between dried plums and prunes in most cases — it's just a new name for a classic product. Prunes have become associated with achieving regularity in bowel movements, and in some cases, the have a reputation as a food frequently served to the elderly in convalescent or rest homes. They have also been marketed specifically for their use as a “health food” for older people.
In order to appeal to a younger market, the California Prune Board began to pressure the Food and Drug Administration to change the name of prunes to dried plums. The name change became official in 2000, so any products marketed with this name in the US are actually prunes with a new name. They are usually of the European type, called the d’Agen plum, the European or the Italian plum.
Plums of the Japanese type are usually eaten fresh, but if they are dried, they are dried plums, not prunes. This has nearly always been the case. The dried prunes from Japanese plums are quite different in taste from what most people in the US think of as plums. They are usually not partially reconstituted, as are prunes, after the drying process.
The name change is only relevant in the US at this point, in part because the European diet tends to welcome and include prunes. The fruit does not have the same connotations in Europe as it does in the US that would make them unappealing to younger consumers, so there's no need for a change.
A similar renaming helped change the way customers perceive the Chinese gooseberry. Once it was renamed as a kiwifruit, sales went up, and people responded to the now popular fruit. The name change for prunes has also led to increased sales among younger people in the US. So, if Shakespeare wondered “what’s in a name?”, it appears he could have been answered that dried plums may taste as sweet as prunes but are more popular.