What is the intuition behind "hardness of approximation"?
One important reason why problems that look equally hard to compute exactly might be very different to approximate relates to the fragility of NP-completeness reductions. 

The simplest example I can think of the relation between VERTEX COVER and MAX INDEPENDENT SET. The two problems are complementary: given a graph $G = (V, E)$ with a vertex cover $S$, the set $V\setminus S$ is an independent set. This is why it's easy to show that one problem is NP-hard by reduction from the other. 

But suppose I can get "close" to a good vertex cover, in that instead of finding the smallest cover $S$, I find a worse cover $S'$ where $|S'| = 2 |S|$. This means that instead of finding the maximum independent set of size $n - |S|$, I've found one of size $n - 2|S|$. 

Suppose in this graph the max independent set size was $n/2$. Then essentially my approximation to the vertex cover found nothing at all. 

In other words, the NP-hardness reduction only preserved the exact solution relationship, not the approximate solution relationship. 

To preserve approximations, you need a different kind of reduction called an $L$-reduction, and once you limit yourself to such reductions, you realize that the two problems are indeed quite different.