Standard example where Jacobson radical does not equal the nilradical

For any local (meaning "has a unique maximal ideal") domain $D$ with unique maximal ideal $M\neq 0$, the Jacobson radical is the maximal ideal $M$, but its nilradical is obviously $0$.

You can easily produce such local domains as localizations of domains at well-chosen prime ideals.

Or, if you're a big fan of valuation domains, one of those would also serve your purposes. A concrete example of this would be $\Bbb F[[x]]$ for any field $F$, whose maximal ideal is $(x)$.

If you want to keep looking for more examples, you might use the following approach. Since the Jacobson radical is the intersection of maximal ideals, and the nilradical is the intersection of prime ideals, you need to look at rings where there you understand the prime ideals and maximal ideals well.

For Artinian rings, the maximal ideals and prime ideals coincide, so it's fruitless to search there.

The formal power series ring $A[[x]]$, where $A$ is any Noetherian ring, is such an example. You can show that $f = \sum_{n \geq 0}a_nx^n$ is in the Jacobson radical if and only if $a_0$ is in the Jacobson radical of $A$, thus $x$ is in the Jacobson radical, but is not nilpotent.