How is NIST's Key Wrap (using AES GCM) different than normal AES GCM Authenticated encryption? From what I could gather, it seems like that NIST's key wrap (publication) provides similar security like AES-GCM Authenticated encryption. If this is true, why did we need a separate specification for key wrap in the first place ?
You can find some info in section B.2. "Comparison of Functionality with Other Authentication Methods" in NIST SP 800-38F, where it said: 


  2) Many authenticated encryption algorithms ... provide an efficient means of authenticating associated data that is not confidential ... 
  
  3) Digital signatures on ciphertexts or MACs on ciphertext can be verified without decrypting the ciphertext. ...


So obviously, NIST KWs don't allow authentication without decryption or associated header data. 

But apart from NIST, you may also be interested in the paper Deterministic Authenticated-Encryption, where it gave a treatment on what does it mean to be a "key-wrapping" mode. To summarize, we can look at the following quote from the introduction section: 


  We call the goal deterministic authenticated-encryption (DAE) ... In a DAE scheme, encryption deterministically turns a key, a header, and a message into a ciphertext. 


Where as traditional AEAD as treated semi-conanically in RFC5116: An Interface and Algorithms for Authenticated Encryption, a nonce/IV needed to be provided so as to provide so called "semantic security". 

As to 


  Q: why did we need a separate specification for key wrap in the first place? 


Determinism decreases the likelihood that misuse of the schemes happens, as could with non-deterministic schemes running on top of broken RNGs or compromised state.