It's common knowledge that bodybuilders need rest after a workout to rebuild muscle. Everyone advises localized rest. That is, you should give a particular muscle at least 48 hours of rest before working that muscle out again. A lesser touched upon subject is holistic rest. This is resting of the entire body - not doing anything for an entire day. What are the endocrinological benefits of holistic rest over localized rest, if any?
A recent article on Juggernaut Training Systems discussing fatigue sheds some light on the subject.
The core problem here is accumulating fatigue. You can arrange your training so that fatigue doesn't accumulate past a certain point; however, most inexperienced trainees don't know enough about themselves to know how to adjust their training. This is a big reason why beginner programs recommend whole body rest between training sessions.
The article doesn't differentiate between localized rest and whole body rest, and at some level your body doesn't either. The biggest danger with localized rest is that some muscle groups are difficult to avoid. For example, it is easy to overuse your triceps and biceps as you work your upper back, shoulders, and chest. It can be done, but you do have to be smart about it. Pay attention to your body, and if your muscles just don't want to work right, those muscles may very well be over fatigued.
There is a very important aspect of your question. To answer it we must consider what happens during exercise.
When exercising a muscle, its energy supplies become depleted, it becomes fatigued and creates by-products such as lactic acid which go into the blood stream. All of these changes stress the body, in response to which the adrenal glands produce cortisol. Despite common belief, and the fact that it is a steroid, cortisol is a catabolic hormone; it suppresses protein metabolism.
How does this then relate to rest periods?
When doing an upper body/lower body regime on alternating days, each individual muscle will get a full 48h of rest. Which is adequate for regenerating the energy supplies within the muscle. However, the body will still be in stress from the previous workout, with increased blood levels of cortisol. By maintaining this sort of protocol, you eventually risk becoming overtrained (depending on training intensity and volume).
However, when you have a full day rest between two trainings, the levels of cortisol and other stress hormones falls, the body regenerates and the risk of overtraining reduces significantly (if this holistic approach is followed regularly).
For an optimal training schedule, you must weigh the benefits of a high frequency of trainings (such as increased strength, endurance and hypertrophy) with the risks (overtraining, fatigue and injury) in order to find an optimal balance between resting and exercising.
Also, as some other answers to this question have stated, there is localized fatigue in stabilizing muscles, such as the core, with any type of exercise. So a full day rest is often needed. The best way to utilize this knowledge is to do block periodization. That is, to exercise on alternate days for a couple of days, say 3-4, and then to take a whole day off. This way you stress the body in waves, but also allow it to adapt and regenerate on the resting days.
The reason you give your muscles that 48 hour rest period is to ensure it has enough time to repair the micro-tears that strength training creates. I know that a lot of people swear by groupings for alternating days (back/biceps Monday, Chest/triceps Tuesday, etc.); however, I find that it is much easier to alternate hemispheres of your body (Upper body on Monday, lower on Tuesday). The reason for this is because: even if you have perfect lifting form, you still use varying muscles as support/stabilizing musculature for lifts. Example, try using a French bar for standing, unsupported bicep curls and NOT letting your pec muscles activate. While you are not actively using the muscles, if your goal is to give your body the best total rest it can have, you should try to not use them as much as possible. Yes, some leg exercises require upper bod stabilizers. However, you use them far less than if you try to split the upper body into several alternating day exercises.
I don't know if you will be able to find a definitive answer to your question, as is the case with many questions in the physical fitness world. There are many outside factors that come into play(mental and environmental stresses that we have no control over just to name a couple) that make it really hard to isolate the impact of just one factor such as rest.
My biggest piece of advice would be to just experiment around with the amount of rest days you take, but spend an appreciable amount of time during each phase of the experiment. Also not that what may be optimal in one phase of your life may not be optimal in another phase(If you just start a new, stressful job, you may have to incorporate additional rest days, for instance).