Almost all words in all languages have more than just one meaning. Some words even have more than 30 different meanings. The verb »abhören« has at least 4 different meanings, they all include a reaction or consequences on what has been heard while »zuhören« does not include reactions or consequences:

• Listening to someone without them realizing it; eavesdropping
  Typical usage: Police suspect some people of committing a crime and have now tapped the phones of members of that group to learn if the suspects are talking about the crime. Here is »zuhören« the wrong choice, because »zuhören« is just passive listening without any consequences. But »abhören« means, that the communication is observed with possible negative consequences for the communicating persons.

  Die Polizei konnte eine Einbrecherbande ausforschen indem sie die Verdächtigen abgehört hat.
  ^{Police were able to track down a gang of burglars by wiretapping the suspects.}
  Der Hacker hat die Kommunikation der Geschäftsführung abgehört und die Mitschnitte an die Konkurrenzfirma verkauft.
  ^{The hacker intercepted the management's communications and sold the recordings to the rival company.}

• A doctor is listening to the sounds of heart and lung. (to auscultate)
  Typical usage: You go to the doctor and she takes a stethoscope, puts it on your chest and listens for the sounds of your heart and lungs.

  Machen Sie bitte den Oberkörper frei, damit ich sie abhören kann.
  ^{Please free your upper body so that I can auscultate you.}
  Ohne Stethoskop ist es schwierig einen Patienten abzuhören.
  ^{It is difficult to auscultate a patient without a stethoscope.}

• Verify that something someone else has learned is correct
  Typical usage: A student has learned something (e.g. vocabulary) and to check if she has learned it correctly, she asks her mother to check it. The mother asks several words in one language and the student answers with the words in the other language. The mother listens carefully to see if the answers are correct. The action that the mother performs is »abhören« in German. The verb zuhören (to listen) would be wrong here, because »zuhören« means just to listen without any reaction. But »abhören« includes giving feedback.

  Kannst du mir bitte die Französischvokabeln abhören?
  ^{Can you please listen to my French vocabulary?}
  Josef hört seinem Sohn das kleine Einmaleins ab.
  ^{Joseph listens to his son's multiplication tables.}

• Verify that a recording sounds well
  Typical usage: A band has recorded a new song in the studio, and after the recording is finished, the sound engineer plays the recording for them, and everyone present judges whether the recording was good or bad. »Zuhören« would mean just to consume the music. But »anhören« includes also a strong critical and verifying part.

  Simon fand bei jeder Aufnahme, die er abhörte, etwas auszusetzen.
  ^{Simon found fault with every recording he listened to.}
  Sollte das nicht auch Thomas abhören bevor wir es veröffentlichen?
  ^{Shouldn't Thomas be listening to this before we publish it?}

Almost all words in all languages have more than just one meaning. Some words even have more than 30 different meanings. The verb »abhören« has at least 4 different meanings, they all include a reaction or consequences on what has been heard while »zuhören« does not include reactions or consequences:

• Listening to someone without them realizing it; eavesdropping
  Typical usage: Police suspect some people of committing a crime and have now tapped the phones of members of that group to learn if the suspects are talking about the crime. Here is »zuhören« the wrong choice, because »zuhören« is just passive listening without any consequences. But »abhören« means, that the communication is observed with possible negative consequences for the communicating persons.

  Die Polizei konnte eine Einbrecherbande ausforschen indem sie die Verdächtigen abgehört hat.
  ^{Police were able to track down a gang of burglars by wiretapping the suspects.}
  Der Hacker hat die Kommunikation der Geschäftsführung abgehört und die Mitschnitte an die Konkurrenzfirma verkauft.
  ^{The hacker intercepted the management's communications and sold the recordings to the rival company.}

• A doctor is listening to the sounds of heart and lung. (to auscultate)
  Typical usage: You go to the doctor and she takes a stethoscope, puts it on your chest and listens for the sounds of your heart and lungs.

  Machen Sie bitte den Oberkörper frei, damit ich sie abhören kann.
  ^{Please free your upper body so that I can auscultate you.}
  Ohne Stethoskop ist es schwierig einen Patienten abzuhören.
  ^{It is difficult to auscultate a patient without a stethoscope.}

• Verify that something someone else has learned is correct
  Typical usage: A student has learned something (e.g. vocabulary) and to check if she has learned it correctly, she asks her mother to check it. The mother asks several words in one language and the student answers with the words in the other language. The mother listens carefully to see if the answers are correct. The action that the mother performs is »abhören« in German. The verb zuhören (to listen) would be wrong here, because »zuhören« means just to listen without any reaction. But »abhören« includes giving feedback.

  Kannst du mir bitte die Französischvokabeln abhören?
  ^{Can you please listen to my French vocabulary?}
  Josef hört seinem Sohn das kleine Einmaleins ab.
  ^{Joseph listens to his son's multiplication tables.}

• Verify that a recording sounds well
  Typical usage: A band has recorded a new song in the studio, and after the recording is finished, the sound engineer plays the recording for them, and everyone present judges whether the recording was good or bad. »Zuhören« would mean just to consume the music. But »abhören« includes also a strong critical and verifying part.

  Simon fand bei jeder Aufnahme, die er abhörte, etwas auszusetzen.
  ^{Simon found fault with every recording he listened to.}
  Sollte das nicht auch Thomas abhören bevor wir es veröffentlichen?
  ^{Shouldn't Thomas be listening to this before we publish it?}