EDIT: I have expanded this answer, after having a few more time.
When Wright introduced the Fst index in 1951, there were no techniques to study the "genetic" components of a trait, and the whole field was based on the observation of the phenotypes. All the studies were based on counting characters that could be observed externally, like the height of the plant or the color of a flower. Population genetics was a theoretical field, and none of the theories proposed by Wright could be verified at the genetic level.
This changed dramatically with the advent of protein electrophoresis in the 1970s. Electrophoresis allowed, for the first time, to observe the underlying genetic component of a trait. Elecrophoresis allowed to determine how many protein isoforms produced the same phenotype, and how many isoforms were present in a population. This introduced a drastic change in the field of population genetics, because it was the first time that many principles theorized up to then could be verified - and some of these principles were found to be in contrast with what was observed in population genetics. The most important change is Kimura's neutral theory - Kimura observed that the number of protein polimorphisms observed were too high to be explained by darwinian selection, and proposed that evolution was governed by genetic drift.
So, the main difference between Wright's and C-W's Fst is the advent of protein electrophoresis. C-W's Fst was adapted to determine the Fst using data from protein elecrophoresis. As Tiago Antao said, Wright's Fst is a theoretical index, while C-W's Fst is a estimator of the former.
Before the advent of electrophoresis, the common view was that most loci were bi-allelic. For historical reasons, it was believed that each gene could have only two alleles, a-g. A and a, as in Mendel's study. Wright himself developed a framework to study multi-allele loci (the infinite-loci model), but this was not used until the 70s and Kimura.
Thus, W-C introduced a way to estimate Fst on multi-allelic loci. When they published the paper in 1984, the existence of multi-allelic loci was acknowledged, as electrophoresis had demonstrated that a protein could have more than one isoform.
In their 1984 paper, C-W dedicated a section to the problem of sample size. I guess that this is due to the need to calculate the Fst index on real data, from protein electrophoresis. Probably one of the first issue at the time was to compare two or more populations, using different sample size.
Moreover, in the 1984 they were in the middle of the debate about the Neutral/Nearly Neutral Theory evolution. In particular, Tomoko Ohta proposed that the strength of genetic drift depends on the effective sample size of a population - if the effective sample size of a population is large enough, then the effects of genetic drift are lower. Thus, the concept of effective sample size was very important at the moment, and in the middle of the debate - so I imagine that it was important to determine if the number of samples in a study were enough to determine the genetics of the whole population.
Formulas (previous answer)
As written in the same 1984 article, there were multiple definition of the Wright's Fst statistics. This was the most common:
While Cockerham's Fst is: