Cork can be shaped easily without any special tools. Lathes and other tools make the job go faster and will undoubtedly produce more consistent results, but you don't need them to produce a well built, comfortable handle. If your handle is old and stained and full of nasty stuff such as oils, perspiration, fish attractants, or any other manner of other gunk that can easily contaminate it during the course of normal use, you will find that water on the handle will make it slippery and difficult to hold. Junk cork is a pain to deal with. As with any material that comes from nature, cork can have inconsistencies inherent in it that can create problems down the road, either during assembly or during use.
Cork is more expensive than foam.
For some, synthetics are the materials that they'd rather have their handles made of. Older materials used in synthetic handles had a bad habit of hardening or breaking down after lots of use, or UV exposure. The newer materials are supposed to be much better than in the past. Synthetics are generally very homogenous throughout. Since they are man made materials, quality control during manufacturing insures the quality of the final product.