By dampening the string vibrations, vibration dampeners change the sound of the ball impact. In particular, the "singing" typical of monofilament strings is significantly reduced. That is probably the most influential aspect of using a vibration dampener. In this case, the reason for using it is mental rather than physical.
It is rather unlikely that tennis elbow can be avoided by using vibration dampers because they do not reduce the vibrations oft he racquet´s frame.
Some players do not use vibration dampers because this allows them to feel and hear more precisely whether they really hit the ball perfectly.
A side effect of the vibration damper is that the racquet's balance point shifts. It makes the racquet a little more head-heavy.
The majority of the top tennis players in the world do use vibration dampeners. On the women's WTA Tour they are even used by around 75%.
Important Note: Before you can check the condition of your strings with the mobile stringster app, you have to remove the vibration damper!