I guess there is the following difference between ordinal and cardinal numbers: while zero is a cardinal number, there is no ordinal number that corresponds to it: it makes no sense to talk about a (or the) "zeroth" something. Curiously enough, I think that there are many occasions where it is meaningful to talk about negative ordinal numbers. If I am considering a sequence of weeks, for instance, and only the weeks after some moment have some relevant feature, it will probably be reasonable to number those weeks with positive ordinals and to numer the previous weeks with negative ordinals. What do you think?
In physics there is the Zeroth Law of Thermodynamics, a transitivity of temperatures. If A and B have the same temperature, and B and C have the same temperature, then A and C have the same temperature. (A could be melting ice, B expanding mercury, and C bourbon at the correct temperature.) The name for the Law was devised after the names for the three laws of thermodynamics had become standard.