These are, of course, different types of studies with different advantages.
A cross sectional study is one that looks at a large group of subjects at one point in time. It might look, for example, at the rate of criminal activity among many different people at one time. The major advantage of this type of study is that it can look at different types of people and it can be done all at once.
A longitudinal study follows the same (smaller) group of people over a longer period of time. This has the advantage of being able to see what happens to specific individuals as they go through life. It can look, for example, at whether people who are involved in crime as juveniles tend to commit more crimes as they get older. This gives researchers a much better feel for what factors can affect outcomes in the lives of particular people.