The degree of importance of the symbolic method in mathematics is unclear. Some have hailed it as the force behind the most effective work in invariant theory, while others think it is grossly overrated. Regardless, the method has stood the test of time as even the lowest dimensional cases of the method are finding uses today.
The most difficult step when using the symbolic method is called the restitution step. This step involves a mysterious variable that never appears alone; one of the creators of the method believed it may be incapable of doing so. This step is not always necessary and can be skipped when solving simpler equations. Another mysterious aspect is the notation it uses for invariants, which is compact and depends on the introduction of new symbols with contradicting properties.
The symbolic method can be used to solve equations such as linear equations and combinatoric problems. Combinatorics is the branch of math that studies the mathematical properties of counting, arrangement and derangement of objects. In problems involving combinatorics, the symbolic method works to translate the relations between combinatorial classes into equations in the respective generating functions. You can then find the asymptotics of the function's coefficients, which provide the desired statistic, by treating the functions by singularity analysis.
Soon after the symbolic method was first created, Cayley and Sylvester recognized that it was not foolproof. A possible problem lay in the need for close examination of the generated invariants and covariants, which was necessary so that the reducible ones could be discarded. If the reducible invariants and covariants were not identified and instead combined with the irreducible ones, they had the potential to cause you to overestimate the number of invariants and covariants, which would throw off the system.