Rolling take off compared to take off from a standstill
There are a few things that may be going on. 

Generally the foot on the breaks, full throttle, spool up, release and roll is derivative of a short field takeoff. In other words you need all available runway and all available power to get off the ground safely and clear your runway end obstacles. This is dictated by airplane performance and runway conditions/length.  

Roll on, roll into full throttle and go is more or less standard. In some cases this may follow an ATC clearance like 


  123AB Cleared for takeoff 22 depart no delay 


This generally means there is either someone inbound for a landing but theres time for you to go, or there is a long queue on the ground and the controller wants you to get going. Stopping on the runway to spool up your engines takes time and at a busy field that can be in short supply.  

Its important to note that in any case the pilot should line the aircraft up with centerline prior to applying full throttle. 

The aircraft can also be instructed to, "line up and wait" in which case its cleared to enter the runway and wait on the numbers until cleared. This may occur at an airport that offers parallel runway operations and the aircraft is waiting for a plane to land on a parallel runway. 



EDIT: I found a possible dupe, noted in the comments. Its a bit different so ill leave this answer here and let the community decide.