Post by Nulgar
Because Fantasy.
In Fantasy stories, real-world myths (or previous Fantasy universes) are usually only taken as inspiration, certain motifs that are morphed into what the author intends - sometimes the name is kept despite substantial differences to help the reader/viewer get an instant general grasp of the topic, sometimes altogether different names are chosen.
WoW's shamans and druids are indeed quite different from their real-world namesakes. For a start, I recommend the Wowpedia articles
Shaman and
Druid, especially their respective "History" sections.
In a nutshell:
Shamans call to the "lifeless" elements of nature - which are in fact quite conscious in this universe - as well as the spirits of the ancestors, and act as ambassadors between these and life / the mortal races.
Druids, on the other hand, stand for and have a grand connection to the "life" in nature, the plants and animals, promoting and protecting its growth.
There are of course many overlaps between the two classes - as I mentioned, the elements are technically "life" as well in Warcraft, as shown by the
elemental allies and
spirit animals that a shaman can summon, while druids can
control the elements to some extent, too.
And that's just considering ingame mechanics, there's much more in the lore regarding their powers and motivations.