Over long distances, there are very few animals that the human (in proper condition) cannot outpace. Horses, however, are no slouches themselves at long distances. I would say you probably get about the same rates on foot or by horse, assuming conditions do not hinder the horse unduly (you'll never get a horse up Everest and a man can fit through more closely grown trees). The great advantage of mounted travel is that somebody else is doing the work. You also generally use a beast of burden, which can handle more weight than a man.
Dean Karnazes "Ran 3,000 miles (4,800 km) across the United States from Disneyland to New York City in 75 days, running 40 to 50 miles per day, 2011" per Wikipedia. (http://en.wikipedia.org/wiki/Dean_Karnazes)
Armies will travel much slower, due to the amount of equipment required, including considerations of food for such large groups. Armies generally include wagons, which will always be slower than simply and horse and rider. Smaller bands of travelers who are less burdened will go faster. Maximum rates require that one be essentially unburdened; running over 100 miles in a day is simply not done with a backpack.
I'd say 20-40 miles per day on a good road and no hampering conditions is about what you can expect with a normal group either on foot or mounted. Conditions will often make travel slower. Don't forget that weather will have an effect; travel can be slowed considerably if the skies are unfriendly. The need to obtain food is more of a problem with horses, if there is no ready grazing. How elaborate the encampment preparations are will also have an affect on travel rate, as will daylight hours, as both influence hours spent in travel. An army that fortifies its encampment will be more secure, but travel slower, than one that simply throws down its bedrolls, sets a watch, and sleeps.
And as fifty's post points out, there is a difference between how far you will travel in a day if you simply need to get where you are going and how fast you go if you need to reserve enough strength to do battle or otherwise exert yourself when done with the journey. Peaceful pilgrims in a peaceful land will outpace an army or a group of "adventurers" exploring wilderness. There are many factors, which is why historical numbers vary so much.
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