The answer is nearly as simple as “for the same reason that you have a nose and a mouth” – the trunk is a bit like an extended nose (and top lip), and noses and mouths do quite different things. The mouth of an elephant is mostly for chewing and swallowing. It contains the teeth and the tusks if they have them (these are incisor teeth that grow so long that they stick out of the mouth either side of the trunk), and also the tongue. The trunk, on the other hand, is used for lifting, grasping, smelling, throwing dust over themselves, smacking baby elephants, making a noise, creating a shower, greeting other elephants (entwining and caressing) and of course breathing as the nostrils run all the way up inside it.
When eating and drinking, the trunk and mouth are both used, but for different purposes. The trunk selects and prepares the mouthful, and then passes it to the mouth for chewing and swallowing. Likewise drinking. Elephants do not drink through their trunks, any more than you could drink through your nose. Instead, the trunk sucks up the water (holding up to perhaps ten litres) and then blows the water into the mouth for swallowing. So elephants have a trunk and a mouth because they need both.
Simon Garrett, Head of Learning, Bristol Zoo Gardens