Most people opt to mulch. Not only does mulch help to control the weeds, but if you select a natural mulch it also adds organic matter to the soil which makes for better gardening results down the road. The newspaper will block the sunlight from reaching the surface of the soil and help to keep weed growth to a minimum. The newspaper will eventually decompose, and not permanently alter the make up of your garden. For one, neither one of them ever go away, and the make-up of your garden is forever altered until you physically remove them, which is a real pain in the butt.
Plastic is no good for the soil because soil needs to breath.
Plastic blocks the transfer of water and oxygen, and eventually your soil will suffer as will your garden. It's all right to use plastic in a vegetable garden as long as you remove it at the end of the season and give the soil a chance to breath. Weed barrier fabrics allow the soil to breath, but what happens is that when you mulch over top of the fabric, which you should because the fabric is ugly, the mulch decomposes and becomes topsoil. Weeds love topsoil, and they will grow like crazy in it. The problem is, they are growing on top of the fabric, and you are stuck with a weedy garden, and a major job of trying to remove the fabric that is now firmly anchored in place because the weeds have rooted through it.