what is the difference between connecting a case fan to fan headers of the motherboard AND connecting it to Molex connector?

Solution
the molex connector will power the fan and normally run it at 100%. the mobo header will normally allow for you to control the fans speed and to set a fan curve if you wish through software. though this usually requires the 4 pin connection rather than the older 3 pin mobo connection.

obviously the fan at 100% gives the most airflow, but also creates the most noise and lowers the life of the fan some. a 4 pin connection and setting a fan curve is the better way to do it. this will adjust the speed of the fan based on temps. so you'll only get high speeds when needed.
the molex connector will power the fan and normally run it at 100%. the mobo header will normally allow for you to control the fans speed and to set a fan curve if you wish through software. though this usually requires the 4 pin connection rather than the older 3 pin mobo connection.

obviously the fan at 100% gives the most airflow, but also creates the most noise and lowers the life of the fan some. a 4 pin connection and setting a fan curve is the better way to do it. this will adjust the speed of the fan based on temps. so you'll only get high speeds when needed.
 
Solution