Layer style = a processing that is applied to a layer before it is shown. For example the drop shadow. NOTE: the original layer content stays intact, but hidden. Only the affected content is shown. This makes possible to remove the effect or to adjust its parameters
Adjustment layer = a processing receipe that is applied to the underlying single or multilayer image before the image is shown. For example you can add a brightness and contrast adjustment layer. NOTE: the adjustment layer do not affect permanently until merged with underlying layers. This makes possible to adjust the effect parameters or to remove the effect. Adjustment layer can have a layer mask.
The difference: Layer style must be assigned to a layer. The style is not itself an item in the layer stack. Adjustment layer appears as an item in the layer stack and affects the image that lies under the adjustment layer. The set of possible adjustment layers is different than the possible layer styles.
ADDENDUM due the comment:
You can't create the same effect as a layer style and as an adjustment layer. See the lists, what is selectable. For example there exists no layer style "Brightness and Contrast" and there's no adjustment layer that creates drop shadows. Why? The reason is unknown. I'm ready to rebuy my copy of Photoshop, if Filters and layer styles were available as adjustment layers with layer masks.
Filters are destructive processes that affect only to the current layer which you have selected to be under your work just now. That layer must contain some bitmap image, not paths or texts or adjustments or smart objects. You can't define a mask to reduce the filtered area or the depth of the filtering. But you can make a marquee selection that defines the affected area. (layer style applies to the whole layer)
I recommend to make a spare duplicate of the layer before applying a filter. A save is also a good idea, because some filters are so heavy loads that the computer can get stucked.