What is the difference between order of base point and curve order in EC?
What's difference between n &amp; #E(FP)?


The difference is that $n$ is the smallest positive integer where $nG = O$; while you correctly state that $\#E \cdot G = O$, that doesn't mean that $\#E$ is the smallest integer that makes this happen.  There may be a smaller integer $n$; $n$ will always be a factor of $\#E$, however it can be smaller.

As for why we would want to make it smaller, that is, why would we want to have an $h = \#E/n &gt; 1$, well, that's exactly what this question addresses.