Similar to the causality Wolfgang mentioned in his post, we could see the reason why time dimension is special from Minkowski spacetime point of view:$ \newcommand{\rd}{\mathrm{d}} $
The $(3+1)$-dimensional spacetime has an inner product defined as $$ (A,B) = A_x B_x + A_y B_y + A_z B_z - \dfrac{1}{c^2}A_t B_t $$ if $A$ and $B$ are two 1-form in Minkowski spacetime: $A = A_x \rd x + A_y \rd y + A_z \rd z + A_t \rd t$, $B$ is defined in a similar fashion, the intuition behind defining an inner product(or rather to say, metric) is to imposing the idea of absolute light speed, such that two different points(events) in the spacetime have zero distance(happens at the "same time", like we are observing the motion of galaxies billions of lightyears away as if they are moving right now) if they are on the same light cone.
As you can see, this inner product is not positive definite due to the presence of the time dimensional scaled by the light speed $c$, therefore intuitively speaking, when treating a problem concerning a quantity propagating in the spacetime, we cannot simply apply theorems in 3-dimensional Euclidean metric to a $(3+1)$-dimensional spacetime, just think of 3-dimensional elliptic PDE theories and their corresponding numerical methods differ drastically from the hyperbolic PDE theories.
Maybe off-topic, but another major difference of space vs spacetime(elliptic vs hyperbolic) is that most elliptic equations model the equilibrium and ellipticity gives us "nice" regularity, while there are all kinds of discontinuities in hyperbolic problems(shock, rarefaction, etc).