The reason for this is that it is not important to actually catch Montag himself. What is important is for it to APPEAR that Montag has been caught. In addition, they (the police) need to catch him right away. These are the reasons why they do not pursue him further.
Catching Montag is only important so that the people will know that criminals are always caught. So as long as someone is caught, it doesn't matter if it's the realy Montag.
Since people have such short attention spans, it is necessary to catch "Montag" right away.
So, it is better to catch a fake "Montag" right away. Catching the real one a week later would do no good.