What is the difference between Big O and Theta notation in terms of inputs? In Coreman , it's written :


  The $O(n^2)$ bound on worst-case running time of insertion sort also applies to its running time on every input. The $\Theta(n^2)$ bound on the worst-case running time of insertion sort,however, does not imply $\Theta(n^2)$ bound on the running time of insertion sort on every input.


So, is there any difference between the two notations in terms of input? If so, then why does it use $\Theta(n^2)$ in worst case running time of insertion sort?
The best case runtime of insertion sort is linear, i.e. $\Theta(n)$. Therefore, there is no $\Theta(n^2)$ bound on all inputs, since that would require $\Omega(n^2)$ on all inputs.

This is just one example proving that while $O$-bounds on the worst-case translate to all-case, but $\Omega$-bounds (and therewith $\Theta$-bounds) do not. (Which is, of course, the point in the text.) You can also investigate this mathematically by unfolding the definitions.