Both synthetic paper and felt paper are an additional underlayment to ice and water shield. They do not go on top of ice and water shield but instead they go on all roof areas not covered by ice and water shield so essentially the remaining area of your roof. Their functionality is the same....to help the shingles breath and prevent the wood deck portion of the roof from absorbing any tar or moisture from the shingles themselves as well as to provide a smooth and flat surface for the shingles to lay on top of.
Tar or felt paper is the traditional and "old school" underlayment used under your shingles. It is a good product and provides some water resistant properties should you have shingles come off and leaving the roof exposed. It can break down, crumble and even rot in time so it's not a totally weather proof product. It's not the best choice for roofs with direct southerly exposure or shallower pitch roofs as it will break down faster under the shingles than synthetic paper.
Synthetic paper is the new and improved underlayment system. It is made of a polypropylene material that has much better water resistant properties with some even being waterproof and manufacturers stating that a roof can be left covered in synthetic alone for up to six months. The brands of synthetic we typically use are Stormtite, Roofguard-Cool grey, Deckgard, Suredeck or Rhino. Synthetic paper is about double the cost of felt or tar paper. The installation method and labour required is the same so overall cost difference to your estimate between the two adds about $200 to the average roof if choosing synthetic over felt. Roofing for Less' opinion and recommendation is once you have decided you want an underlayment then spend the extra money on the synthetic paper and take the upgrade. We rarely recommend felt or tar paper and only install it if tight budget constraints are present. Synthetic paper is also lighter in weight than tar or felt paper.
In some cases an underlayment is mandatory. If you have a board roof then you must have an underlayment of some sort to create a smooth surface and to protect against moisture under shingles which is more likely with a board roof than a plywood covered roof. If you want the extended warranties that the shingle manufacturers offer then synthetic paper is mandatory - felt or tar paper does not qualify. If you have a shallower pitched roof 4/12 to 5/12 then you should have an underlayment. Any roof with a pitch 2.5/12 up to 4/12 should have ice and water shield. Any roof below a 2/12 should have a flat roofing application.
Underlayments were also absolutely necessary with the older organic asphalt shingles. With the recent shift to sole production of the laminate fibreglass shingles with most manufacturers you can get away without an underlayment if cost is the main priority. The laminate fibreglass shingles do not contain the same amount of tar and other organic properties that the asphalt shingles once did so there is less chance of moisture retention and tar absorption onto the wood deck surface.
Synthetic paper or felt/tar paper is never a bad option and in all situations serves to preserve and extend the life and improve the appearance of your roof.