I do not really think that this model has a great deal of usefulness, at least in the short run.
In the short run, I believe that wages are "sticky" or "rigid." The competitive labor market model assumes that wages will rise and fall easily as the supply and demand of labor changes. However, because of union contracts and other such factors, I do not believe that wages change so easily. Please follow the links for more discussion of this kind of factor.
However, the model is generally useful for explaining why workers in some jobs earn more money than others (because there is less supply relative to the demand in the higher-paid jobs). It can also help us understand why wages in some countries are higher than in others.
Overall, then, I would say that it is not very useful in the short run in individual labor markets but it is useful in the long term and in comparing labor markets.