What is the difference between the Pearson correlation coefficient and the Spearman correlation?

Pearson’s product-moment correlation is used on two continuous variables that are not non-linearly related; Spearman’s rho is used on ordinal data but also works for monotonic relationships.
Pearson’s product-moment correlation is employed when you have two interval or ratio (i.e. continuous) variables. In these cases the difference between scores is meaningful (i.e. the difference between 4 and 6 is equivalent to the difference between 6 and 8). Pearson’s product-moment correlation requires that the two variables are not non-linearly related (i.e. they can have no relation).
Spearman’s rho is employed when the data fail to meet either of the two conditions above (i.e. the data are ordinal or the relationship between variables is non-linear). Spearman’s rho can be used to straighten out a monotonic relationship (i.e. a relationship that always has an instantaneous rate of change that is positive or negative). It will not work on functions whose derivative changes sign across the domain of ##X##.
In terms of calculation there is very little difference. In fact you can use the same formula. However there is a handy computational formula for Spearman’s rho when there are no tied ranks and the sum of the ranks for the two variables are equal.