LP Approximation: Primal relaxation + rounding vs. Dual relaxation. Why is the latter better? Given any Integer Linear Program (ILP) there are 2 ways to approximate it:


Write down ILP, convert to LP by relaxing the integer constraints and round the solution
Write down the ILP, convert to LP by relaxing the integer constraints, write it's dual and solve it


Why is the second approach considered 'better'? What makes it the preferred way of approximating the solution to an ILP? What is that #2 has that #1 doesn't?

In Vazirani's Approximation Algorithms book he approximates set cover using both of the methods above, but I'm unable to discern the underlying concept of choosing #2 over #1? What "IS" the intuitive 'aha' to help me understand this?
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.