Fever is "an elevation of body temperature that exceeds the normal daily variation and occurs in conjunction with an increase in the hypothalamic set point." In other words, in a fever, the body is trying to raise the temperature above normal, as a response to pyrogens (like bacterial antigens). I've covered the details of the interesting fever mechanism here.
On the other hand, hyperthermia is characterised by "an unchanged (normothermic) setting of the thermoregulatory center in conjunction with an uncontrolled increase in body temperature that exceeds the body's ability to lose heat." Causative examples of hyperthermia include heat stroke (e.g. exercise in very hot conditons), drug-induced hyperthermia (e.g. from monoamine oxidase inhibitors), and malignant hyperthermia.
Fascinatingly, the response to antipyretic medications (e.g. paracetamol/acetaminophen) is completely different in hyperthermia, as compared with fever. In the latter case, the drugs successfully lower the hypothalamic set point back towards normal, by interfering with prostaglandin production. In hyperthermia, however, lowering the hypothalamic set point makes no difference, as it isn't what is causing the elevated temperatures. In fact, it is already likely to be at rock bottom, temperature-wise, in response to the high body temperature.
* Harrison's Principles of Internal Medicine (of course), 16th edn.