The key point is: different functions/distributions may induce the same linear functional on a space, acting in different ways. For example, on $H^1(\mathbb{R})$ we have a continuous linear functional $E_0v = v(0)$ (evaluation at $0$). This functional is induced by the function $u(t)=\frac12 e^{-|x|}$ under the action
$$
v\mapsto \int_{\mathbb{R}}(uv+u'v')
$$
On the other hand, the same functional $E_0$ is induced by the distribution $\delta_0$ (Dirac delta) under the action
$$
v\mapsto \int_{\mathbb{R}} \delta_0 v
$$
where the integral isn't really an integral but the action of a distribution.
So: two different distributions, acting in different ways, yield the same result. The Dirac delta is the distribution that yields $v(0)$ when applied to test function $v$. The function $\frac12 e^{-|x|}$ does not do that: when applied to $v$, it yields $\int_{\mathbb{R}} \frac12e^{-|x|}v(x)\,dx$, not $v(0)$.
If we use a different way for distributions to induce linear functionals on $H^1$: namely, integrate also against $v'$, then $\frac12e^{-|x|}$ induces $E_0$ functional, and $\delta_0$ induces some unbounded functional.
Functions like $\frac12 e^{-|x|}$ are called a reproducing kernel for a Hilbert space.
Here's one more way to look at it: the map $u\mapsto u-u''$ is an isomorphism of $H^1_0(\mathbb{R})$ onto $H^{-1}(\mathbb{R})$ which sends $\frac12 e^{-|x|}$ to $\delta_0$. But "isomorphic" does not mean "the same space".
