Does image's background matter for detector training (CNN)?
Of course, it matters.

Which one is best completely depends on your problem. The golden rule for machine learning problems is that you want the data you train on to be as representative as the data the model will encounter once live.

Typically, faces cropped dataset will be an easier problem because the CNN can directly focus on the face, but if your use case will use images of faces that are not cropped, your model might not perform well.

So use whatever data you expect to encounter later on. The only thing you could do is add a face-cropping algorithm before your network, and then you could train on the faces cropped dataset right away.