From what I've read it's important to have a rest day if you are doing very intense work-outs or if you are weight training to build muscle. The day off gives your muscles and ligaments a chance to repair and rebuild. One site I went to said that knee pain as a result of running was often because the ligaments aren't given a chance to recover between days resulting in a form of RSI, not necessarily as a result of prolonged wear. So it seems to be important for all hardcore work-outs, not just weight-training where the rule is traditionally observed. I'd say that with low to medium intensity work-outs you wouldn't need the day off as the muscle wouldn't have degraded as much, though.
The personal trainers on the forum would know better than me, but I'd guess that if your muscles ache the next day it's an indication that they need to recover. If you're feeling fine then I see nothing wrong with more exercise provided it's not too intense. If you're doing intense work-outs, high impact exercise or weight training I'd rest between days. Even alternating different types of exercise on different days should be effective since you would be working different muscle groups in different ways (so one day running, the next day weights, etc.)