Layer Styles vs Effects (example: Drop Shadow)

A raster layer is one that's made up of pixels, as opposed to a vector layer which is created with math. A vector layer is infinitely scalable, but a raster layer has a defined resolution. The render order changes when vector layers are involved, so it was easiest to demonstrate this on a raster layer like an image or a video, as opposed to a native After Effects solid or Shape Layer.

In general I stick with effects because overall there's more flexibility. You can reorder them, mask them, and access them in the Effects Controls panel. Aside from Photoshop compatibility, though, Layer Styles do present some benefits. On a raster (non-vector) layer add a Drop Shadow effect. On a duplicate of that same layer add the Drop Shadow Layer Style. If you scale or rotate the layer with the effect you'll see that the shadow is fixed to the layer, which isn't natural. However, if you scale or rotate the layer with the Layer Style on it you'll see the shadow reacts more naturally, as if there was actually a light cast on the layer creating a real shadow.

Because of the ease of access and flexibility, I would reccomend sticking with effects, but be aware of the different Layer Styles and understand how the rendering order affects these things. Knowing both of these will help you to decide what's the best tool to use. If you want to scale or rotate a layer with a shadow and have it look accurate, or as Adirai mentioned, use the Global Light property, then the Layer Style is best. 

Another Layer Style to look at is Stroke. You can easily outline a raster layer since you can't put a Shape Layer stroke around it. The Stroke Layer Style lets you specifcy "Inside," "Outside," and "Center" for alignment, unlike the Shape Layer Stroke. So you can even use this layer style on a shape layer with vector strokes too!

Yes, there are some pretty significant differences between the two. Layer Styles are actually Photoshop Layer Styles and you can preserve the same settings you chose in PS when importing layers into AE. Another huge difference is in the way After Effects' render order is handled. Here are a couple links that explain this, as well as the render order in After Effects. 

Layer Styles: https://helpx.adobe.com/after-effects/using/blending-modes-layer-styles.html#layer_styles

Render Order: https://helpx.adobe.com/after-effects/using/precomposing-nesting-pre-rendering.html#render_order_and_collapsing_transformations

I can say a difference that, there's global light option in Layer Style Drop Shadow option. It's useful when we use it in precomps.

Just try this you will know the difference Create a text layer Duplicate and make two different precomps. Apply drop shadow effect on one precomp. Apply dropshadow layer style for another precomp & Turn On global light in layer style.. Make the properties value same for both layer style and effects.

Then rotate both layers and see the shadow fall direction.