NoIR version of the camera should be used mostly when you need to illuminate your scene with IR (=near infrared) illumination. These cases are pretty special and if you don't know why would you use NoIR version, just buy the normal one. The cases where one would need NoIR version are as follows:
1) "invisible" illumination in darkness, e.g. for surveillance purposes or for other uses when visible light is not desired. In that case, you would use LED IR illuminator, the same that comes with many "see in the dark" surveillance cameras.
2) Overpowering the undesired stray light coming into the scene. For example, in many machine vision applications one needs strict control of the lighting conditions. In that case you would use LED IR illuminator in combination with the visible light cut-off filter (very dark red/black, see here:
This works quite well when artificial light is the problem, as basically none of the power saving light bulbs emit in the IR spectrum. Bigger problem with incandescent lighting and pretty difficult to mask out the sun.
3) Using special IR sources, like lasers for illumination. In this case you would need to match the filter with the (very narrow) spectrum of the laser.
4) In some cases people like to observe natural environment, illuminated by sunlight in IR (e.g. plants become bright white). In this case you would need to add visible light blocking filter as well.