Note that $2010=10(201)=2(5)(3)(67)$ and that 67 is prime. Therefore, the positive divisors of 2010 are $1,2,3,5,6,10,15,30,67,134,201,335,402,670$, $1005,2010$. Thus, the possible pairs $(a, b)$ with $a b=2010$ and $a>b$ are $(2010,1),(1005,2),(670,3)$, $(402,5),(335,6),(201,10),(134,15),(67,30)$. Of these pairs, the one with the smallest possible value of $a-b$ is $(a, b)=(67,30)$, which gives $a-b=37$.