Peat moss is a natural end product from the decomposition of sphagnum moss during its life cycle in bogs. As its mature parts die off, the peat moss is harvested, and since it is a source of rich organic matter, it is a choice addition to soil. Once it is dried, it behaves like a sponge, absorbing and retaining a lot of moisture since it is a natural product from a wet environment.
Peat moss is also added to soils that have a high pH to balance it, since it is slightly acidic.
Cows are vegetarians, and their digestive tracts serve as live composting machines for the grass and hay they consume. The decomposition of the vegetation releases the nutrients in their feed to allow for easy absorption in their bodies. What doesn't get absorbed gets excreted as manure. This becomes a mix of decomposed nutrients and fiber, the fiber of which can aerate the soil, much like peat moss.
Peat moss is high in fibrous materials that aerate and balance the consistency of the soil, but it lacks a substantial amount of nutrients to enrich the soil with enough plant food to sustain its growth through the season. This is primarily because most of its decomposition has taken place before it was harvested for landscape use.
Cow manure, on the other hand, is just the opposite --- you will find that it has sufficient soil nutrients, but the amount and integrity of its fibrous content may not be enough to completely enhance the soil's texture.
Another vital difference between peat moss and cow manure is renewability. Sphagnum moss decomposes much slower than the rate that peat moss is harvested, thereby threatening its future existence. On the other hand, the rate of cow manure production is directly related to their feed. As long as a cow lives and eats, it will produce manure.
Test your soil prior to adding anything to it, to get a general idea of how to balance your soil. If your soil can benefit from peat moss and cow manure, add both.
Break up the peat moss, and spread it over the surface of your soil. Spread the cow manure in the same way. Turn it into the soil, one section at a time, using a pitchfork, a shovel or a garden tiller. If the soil is difficult to break into, water the area first to make the job easier.
Water the area to start to release and mix the nutrients into the existing soil, once the entire area is turned over. Plant your landscape or garden additions, and watch how well they grow in soil that is nutrient-rich and aerated.