Both slate and shale are fissile. They split along planes in the rock. In a shale these planes are the bedding planes of the rock. These mark out the individual layers of clay that were laid down as sediment. In the slate these planes are mineralised layers oriented in response to a regional stress field during low grade regional metamorphism. These planes may be parallel to the bedding planes, but more likely are not.
Jimbob, I would call a 'shale' with no obvious fissility a claystone. But then I recall many an argument with colleagues over whether a sample was a greyish-green shale, or a greenish-grey shale. And once we'd settled that someone else would say it was gray, not grey.