What anions does the loss of a proton from HOCN and HNCO produce? Are they same or different?
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.