You can get gingivitis. Simple as that, or some other gum disease. Flossing is important.
Poor dental health, in particular, not flossing, can lead to other health issues. One, of course, is loss of teeth. If the gums are not stimulated and the plaque is not removed, the gums can become diseased because of the bacteria causing plaque, and cause the teeth to loosen over time. It is not a pleasant prospect to lose teeth, it is a painful process that occurs over a period of years, usually one tooth at a time. The gums become swollen and painful, they bleed very easily, usually just touching a toothbrush to them will cause them to bleed, and the teeth become more sensitive, making eating and drinking difficult at times. Dentists want this whole unpleasant situation avoided, firstly through regular dental checkups and secondly, by regular flossing.