Proving Linear Time Temporal Logic formula □ ◊ f ⇔ ◊ □ f
You are not missing anything. These expressions are indeed not equivalent.
Assume $f$ in your case is an atomic proposition. Then the computation:
$f,\neg f,(f,\neg f)^\omega$ satisfies $□◊f$, but not $\Diamond \square f$. 

Intuitively, $\Diamond \square f$ means "After a finite prefix, there is always $f$", while $\square \Diamond f$ means: "There are infinitely many $f$s".

By the way, you have some mistake in your semantics. It should be:
$(\sigma,j)\models \Diamond f$: $\exists k[k\ge j\wedge (\sigma,k)\models f]$.