What is the exact difference between NBU media server and client?? Do they work differently?

A client has limited binaries, and can basically only send data to a media server.  It doesn't have direct storage attached.

A media server has all the binaries of the master, so you can run any command from a media server, even if that command acts on the master.  It has storage attached and is the part of a nbu environment that receives the data from the clients and writes it to tape/ disk.

The media server does not contain the Nbdb, this is on the master and contains a lot of config and pay off the images database, the other part of the understands is also on the master, but is just flat files. 

Media server component places the data received client component onto storage media (tape/disk).

Client is a data producer

Media Server is data receiver

...In most cases.