Q-Learning: Target Network vs Double DQN
Ok, it's simple!

Just the "Target network approach":


  
  Select an item from Memory Bank
  Using Target Network, from $S_{t+1}$ determine the index of the best action $A_{t+1}$ and its Q-value
  Do corrections as usual
  


The "double DQN approach"":


  
  Select an item from Memory Bank
  Using Online Network, from $S_{t+1}$ determine the index of the best action $A_{t+1}$. 
  Using Target Network, from $S_{t+1}$ get the Q-value of that action.
  Do corrections as usual using that Q-value
  


As seen from 3, the Online Network gives us the action that it thinks is highest.  However, we let the Target Network have the final say on the score of such an action. 

Were we to let Target Network select a best action, it could have very well selected some other index.