LP Approximation: Primal relaxation + rounding vs. Dual relaxation. Why is the latter better?
If one uses an ILP formulation and its LP relaxation then clearly it does not hurt to look at the dual. In many cases the dual helps to understand/interpret the lower/upper bound that the relaxations give. This can be exploited algorithmically in a direct fashion, or some times indirectly to provide intuition. Several classical combinatorial optimization results are based on min-max results where duality based analysis/interpretation is very direct and useful. 

Making the comment an answer based on Suresh's suggestion.