The first and immediate thing is that the mommies may have trouble with their babies. First, the birthing process will be much messier, with blood and hair and all sorts of ickiness stuck to the backside and back legs. Secondly, babies have an instinct to nurse, but if they grab curls of hair instead of teats, they will not get the colustrum and milk they need.
The second problem is one that will grow with the hair. As the hair gets longer and springtime rains come, the hair will mat and felt and generally get quite unruly. For a texture example, picture dreadlocks. Underneath that matting hair, lice and skin problems are a bigger risk. For the male goats, dripped urine will not only stain the stomach hair, but can actually cause rot of that hair and the skin beneath it.