Distinct MAC and Key for AEAD?
The examples I see in wikipedia show that the same key is used for
  both.


Doing so is fine in most constructions, however "most" is not "all". Also the graphics could be read as "derive a key from the given input key somehow".


  Why would the encryption key be different than the MAC key?


Cryptographically speaking, it's cleaner to use different keys for the different primitives, as it definitely prevents subtle interactions due to the key-reuse and thus makes formal proofs about security claims easier. 

Also there are cases where you actually have to use different keys for the MAC and the encryption scheme, the most well-known example being the combination of CBC and CBC-MAC which allows for modification of (nearly) the whole message on key-reuse.