The difference between the two is something you need to memorize for your instrument knowledge test, but don’t necessarily have to understand. John D. Collins on Oct 28, 2012 in a post on Ask A CFI explains it quite well.
The TC indicator provides both bank rate and turn rate, whereas the TS only provides turn rate. Because the TC is affected simultaneously by turn and yaw, it can be difficult to use it to recover from an upset. The old TS has the advantage that it only indicates turn rate, and if you are partial panel, this can save your life.
Back in the mid 1920’s it was fatal if you flew into a cloud. A pilot named Howard Stark worked out a method using the newly developed Turn and Slip Indicator by Sperry to safely fly in the clouds with this instrument. If an upset occurred, he employed what came to be known as the Stark 1-2-3 method to recover control.
First, stop the turn with the rudder so the turn needle is in the center, second center the ball by using the ailerons to level the wings and third control the dive with use of the airspeed indicator and the stick to control the elevators. This method is foolproof and is still used by many experienced pilots today.
Unfortunately, it doesn’t work with the Turn Coordinator because the indicator shows both rate of turn and rate of bank. There are many old and not so bold pilots who either replace the turn coordinator with a turn and slip indicator or install a spare turn and slip in their aircraft when it is equipped with a turn coordinator that is required for the autopilot operation.
More info at AvWeb as well.