What anions does the loss of a proton from HOCN and HNCO produce? Are they same or different? What anions does the loss of a proton from $\ce{HOCN}$ and $\ce{HNCO}$ produce? Are they same or different?


Once, I had asked my teacher if $\ce{HCN}$ and $\ce{ HNC }$ are different or same compunds, he said they were different and yes, they had different structures. Applying the same logic here, $\ce{OCN-}$ and $\ce{NCO-}$ are different anions because they'll have different structures.
When they are bonded to hydrogen that is $\ce{HOCN}$ $\ce{HNCO}$. The two compounds will be different:



From Wikipedia


  Organic cyanates are called isocyanates when there is a $\ce{C−NCO}$ bond and cyanates when there is a $\ce{C−OCN}$ bond.  


Due to resonance after deprotonation the two anions will be the same.