One can think of a topology on a space $X$ as abstracting all the "stable" information (or "physical measurements") one can say about a state $x$ in $X$ (i.e. the open neighbourhoods of $x$ in $X$).
For instance, consider the real number $\pi$ in ${\bf R}$ (with the usual topology). We can't specify $\pi$ exactly in a stable manner, because we can perturb $\pi$ a little bit and it won't be $\pi$. (In other words, $\{\pi\}$ is not open.) But we can say, for instance, that $3.14 < \pi < 3.15$, and this is a stable piece of information (it is true even if we perturb $\pi$ a little bit). The Hausdorff nature of the real line then lets us demonstrate that two quantities are distinct even if we are only allowed to access them in a stable manner. For instance, $\pi$ and $e$ can be stably shown to be distinct, because we have a stable measurement $3 < \pi < 4$ of $\pi$ and a stable measurement $2 < e < 3$ that are disjoint from each other.
Now we work instead with the Zariski topology. Here, we are not allowed to use the $<$ sign to make stable measurements (we are now in the algebraic world rather than the semi-algebraic world). The only way to make stable measurements, then, is to use the $\neq$ sign (in conjunction with the usual arithmetic operations). For instance, one can say that $\pi$ is not equal to $3$, that $\pi^2$ is not equal to $10$, and so forth. This is of course a much weaker topology. In particular, it is no longer possible to use stable measurements to stably separate $\pi$ from $e$. ($\pi$, of course, does obey the stable measurement $\pi \neq e$, and $e$ obeys the stable measurement $e \neq \pi$, but this does not help, because the stable (i.e. open) sets $\{ x: x \neq e \}$ and $\{ x: x \neq \pi\}$ are not disjoint, and so these stable measurements do not force distinctness. In more standard notation, the Zariski topology is $T_0$ but not Hausdorff.) [This has nothing to do with the transcendental nature of $\pi$ or $e$; one also fails to separate, say, $0$ and $1$, in the Zariski topology.]
[One can also take a measurement-oriented perspective to other aspects of the Zariski topology. Thus, for instance, a set $E$ is Zariski-dense if there is no way to exclude an arbitrary point $x$ from lying in $E$ using only stable measurements of $x$. As the Zariski topology is so weak, this is a fairly weak property; there are a lot of Zariski-dense sets.]
In general, non-Hausdorff topologies are usually extremely weak topologies, in which there are very few stable measurements available and so it is hard to stably separate distinct points from each other. The most extreme case is the trivial topology, in which no non-trivial measurements are available at all.
