The importance of Integrality Gap
Integrality gaps essentially represent the inherent limits of a particular linear or convex relaxation in approximating an integer program.  Generally, if the integrality gap of a particular relaxation is $x$, then any approximation algorithm based on that relaxation cannot hope to do better than an $x$-approximation.  So at the very least, integrality gaps are of interest to algorithm designers since they suggest limitations in certain techniques.  

So why not just come up with another LP relaxation or switch to other techniques and move on?  Linear and convex programming have proven to be central to approximation algorithms; for many problems the integrality gap of a natural LP or SDP formulation is equal to the approximation ratio of the best algorithm as well as the hardness of approximation ratio.  This is just an empirical observation, but it means that proving an integrality gap may suggest much stronger consequences of an improved algorithm or lower bound.

There may be deeper and more rigorous reasons for this phenomenon.  For instance, assuming the unique games conjecture, it is known that the approximation ratio and inapproximability ratio for constraint satisfaction problems is equal to the integrality gap of a simple SDP relaxation (see Optimal Algorithms and Inapproximability Results for Every CSP? by Prasad Raghavendra)

Finally, integrality gaps represent unconditional lower bounds.  Usually, we need to rely on unproven assumptions (e.g. $P \neq NP$) if we want to make any progress in lower bounds, but for restricted models of computation, we can sometimes get away without it (see lecture notes by Luca Trevisan).  Integrality gaps, being purely geometric rather than computational, are one way of getting fairly powerful lower bounds without the baggage of extra assumptions.