The sequence $(a_n\delta_n)_{n\in\mathbb{N}}$, where, for each $n\in\mathbb{N}$, $a_n$ is a complex number and $\delta_n$ the Dirac delta translated of $n$, i.e. $\langle \delta_n,\phi\rangle=\phi(n)$ for every $\phi\in\mathcal{D}$, [*] is always convergent to zero in $\mathcal{D}'$ (because, for every $\phi\in\mathcal{D}$, $\langle a_n\,\delta_n,\phi\rangle=a_n\phi(n)$ is zero for "large" $n$).
However (this is exercise 9.3.10 from Jean-Michel Bony, Course d'analyse) it should be convergent in $\mathcal{S}'$ if and only if the sequence $(a_n)_{n\in\mathbb{N}}$ "has moderate growth", which is: there exist $C>0$ and $p\in\mathbb{N}$ such that $\vert a_n \vert \le C\,(1+n)^p$ for every $n\in\mathbb{N}$.
I'm ok with the if part, but I'm not able to prove the only if part, which (by taking sequences $(a_n)_{n\in\mathbb{N}}$ which have not moderate growth) would provide examples of sequences of distributions which are convergent in $\mathcal{D}'$ but not in $\mathcal{S}'$.
[*] I'm a bit confused about the notation of this object; some texts would denote it as $\tau_n\delta$, other as $\delta\circ\tau_n$; physicists usually denote delta as a "function", so would simply write $\delta(x-n)$. In general, for $T\in\mathcal{D}'$, here we're dealing with the distribution $\mathcal{D}\ni\phi\mapsto\langle T,\phi\circ\tau_{-n}\rangle$, which is the pullback of $T$ by $\tau_{-n}$, the inverse of the diffeomorphism $\tau_n$; maybe it should be denoted more formally as ${\tau_n}^\star T$ or ${\tau_n}^\ast T$, to underline its "definition by duality"?
Take $\phi(x)$ some smooth bump function with support in say $[-1/3,1/3]$ and with $\phi(0)=1$. Given a sequence $(b_n)$, construct the function $h(x)=\sum_n b_n\phi(x-n)$. If the sequence is of rapid decay, i.e., satisfies $$ \forall p,\ \sup_{n}\ (1+n)^p|b_n|<\infty\ , $$ then $h\in\mathcal{S}(\mathbb{R})$. For the limit you are looking for to exist in $\mathcal{S}'$, you need $$ \sup_n|a_nb_n|<\infty $$ for all rapidly decaying sequences $(b_n)$. It's not too hard to see that this can only happen if $(a_n)$ is of temperate growth.