Screw-on style filters

Pros

• Cheaper set up costs, but will cost more if you buy several filters.
• Less likely to be dropped or fall off the camera.
• More portable and easier to carry around.
• Offers a little more protection for the lens.
• Circular polarising filters and UV filters are typically screw on.

Cons

• A given filter only fits one size of lens. So each lens of differing diameters requires their own filter or a step-down ring. More expensive in the long run.
• Must move lens to ensure transition on a ND Grad filter is correctly placed, which may impact image composition.
• Greater risk of vignetting on wide angle lens.
• Harder to use multiple filters without vignetting.

Slot-in style filters

Pros

• Holder allows the filters to be used on different size lens, provided you have the relevant adapter.
• Great versatility.
• Less risk of vignetting on wide angle lens, especially when using the larger size slot filters.
• Quicker to change and adjust.
• ND Grad filters typically use slots.
• Easier to use multiple filters at once. Less chance of vignetting when using multiple filters.

Cons

• More expensive to get set up as you need a holder, adaptor and a filter. Costs less if you buy several filters.
• Greater risk of being dropped or falling off the camera.
• More cumbersome to carry around.

So which is better? It depends on what you needs are. In my case The Doc uses both. Screw-in for circular polarising and colour enhancement. Slot-in filters are mainly use for landscape, especially ND Grad filters. The Doc even uses a combination of both, say a circular polarising with a ND Grad.

Tip: If you mostly do landscape photography, slot-in filters is the way to go.