The Difference Between Slow Motion and Half Speed
Slow motion and half speed both create the same type of effect. But one is an in camera effect and the other is achieved in post production. But how they're created is only the beginning of their differences.
Slow motion is a special effect that is created when the camera records an action at a frame rate that is greater than the play back frame rate. For example, typical video plays back at 30 frames per second which is the same rate that you would record it to play back at normal speed. Now if you recorded 10 seconds at 60 frames per second then when it is played back at 30 fps it will take 20 seconds for those 10 seconds to play back. This is how true slow motion is achieved.
Half speed is achieved in an editing program. It duplicates and adds frames to your video in order to make it play back slower. For example, to turn your 10 second clip into a slow 20 second clip the program would duplicate every frame once. So one frame that played for 1/30th of a second now plays for 1/15th of a second.
Differences in Quality
Half speed does not reproduce true slow motion and is thus inferior in quality. True slow motion records at a slow motion frame rate which means that every frame in play back is different with some motion happening in it as opposed to frames being duplicated. The play back is smoother in true slow motion. If you want a slow motion effect and have a camera that can change its recording frame rates then you should achieve the effect in camera rather than post.