= Solution
Two nonisomorphic <punctual scheme>[punctual schemes] are
$$
\operatorname{Spec}k
\qquad\text{and}\qquad
\operatorname{Spec}k[\varepsilon]/(\varepsilon^2).
$$
Their underlying spaces each have one point, but the second has a nonzero <nilpotent element> and the first is reduced.
After translating their common support to the origin, punctual closed subschemes of $\mathbb A_k^1$ correspond to ideals of $k[t]$ whose radical is $(t)$. Since $k[t]$ is a <principal ideal domain>, each such ideal is $(t^r)$ for a unique $r\geq1$. Its coordinate ring has basis $1,t,\ldots,t^{r-1}$ and hence dimension $r$. Equal dimensions force equal exponents, so $Z$ and $Z'$ are in fact the same closed subscheme after the common coordinate choice, and in particular are isomorphic.
In $\mathbb A_k^2$, the ideals
$$
I=(x,y^2),
\qquad
I'=(x^2,y)
$$
define distinct punctual closed subschemes supported at the origin. Both quotient rings have dimension two over $k$, with bases $1,y$ and $1,x$ respectively. Thus they have equal-dimensional global-section spaces despite being distinct embedded closed subschemes.
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