In choosing whether to go with wool or synthetic carpet, it is adviseable to understand firstly what the advantages and disadvantages of these respective fibres are:

Polypropylene

Of the synthetics, Polypropylene is less expensive than Nylon, stain-resistant and bleach-cleanable. Generally, most cold liquid stains will not stain a polypropylene carpet, although it should be noted that hot liquids present a different problem. If a hot drink for example is spilt, it is possible that the heat of the liquid, can in some circumstances simulate the original dyeing conditions and bond the stain. Also, as you might imagine, the fibre of these types of carpet can melt if subjected to sources of heat. For this reason, it would be unwise to use them in a room where you might have cinders spitting out from an open fire, or possibly where there would be any risk of dropped cigarettes. On the other hand though, if the children drop a glass of Ribena, the stain should come out.

Nylon

Nylon, is a slightly more expensive fibre than polypropylene, and is likewise stain-resistant. For technical reasons however it is not bleach-cleanable. Despite having better abrasion and crush resistance than polypropylene, or even wool in some cases, nylon is a far softer fibre. When manufactured as a tufted carpet, it has a very plush velour feel and is often used in the construction of  'Saxony' or 'Velvet' syle carpets. It is an extremely durable fibre and when constructed as a low-level loop-pile carpet, is the preferred choice for very high traffic areas in contract flooring situations. Like polypropylene it can melt when exposed to a direct heat / flame.

Wool

When it comes to carpet, wool is still the preferred option for many people. Wool has excellent crush resistance, and moderate stain-resistance (although it is possible to improve this with factory or retrospectively applied stain-guard treatments). If not actually burn resistant (hence it's widespread use in pubs before the smoking ban), wool burns slowly  and self-extinguishes in flames (hence it's widespread use in pubs before the smoking ban). The only downsides really are that it can be attacked by carpet moths, and can have poor sunlight fade resistance in certain pigments.