We experience space and time very differently. From the point of view of physics, what fundamentally grounds this difference?
Kathryn, in one sense your question answers itself. It may not be the most satisfying sense, but it's an important one.
The distinction is the one Einstein first made when he proposed special relativity, before his former professor Minkowski re-framed Einstein's theory in more graphical terms: Time is the ticking of a clock, that is, the quantification of measurably cyclic processes. Einstein at first did not attempt to frame this idea into highly graphical terms, but of course did so later after some initial mild grumblings about how Minkowski had made his own theory incomprehensible to him.
It is of course trivially easy to take the concept of regular clock ticks and make that into a concept of distance, but it is not a trivial mapping from the perspective of what must be available to make the mapping. It just seems that way because as organisms capable of existing and surviving in our particular universe, we come pre-equipped with the necessary hardware, and with a situation that makes the idea meaningful.
Does all that sound too complicated for something as simple as counting clock ticks and then representing them along a length-like line? Not really.
You can't access the past as you can a distance, so where does the knowledge of the past reside? In something called a "memory" or a "storage device," which must be independent of the part of the clock that does the ticking. So, you have memory.
You cannot interpret memory without some set of operations that recognize and can act on such a representation of past cycles, treating them within some kind of very different construction as if they existed again. Such operations constitute form of intelligence, including a rather remarkable ability to "simulate" or reproduce the physics of a past event, despite having nothing left of that event except a very wispy pattern of information (itself a very strange concept) about key features of that past event. This collective ability to simulate and operate on wispy, ephemeral, extraordinarily incomplete images of the past, yet somehow manage to achieve a meaningful reproduction of their consequences, we call "intelligence."
But how is that even possible? It seems rather absurd that such slender representations can in fact make meaningful predictions of a much denser and tremendously complex collection of matter an energy.
There the universe itself helps us out both by being based not on total chaos, but on slender, simple, and uniform rules of operation. In the case of time, the universe assists us tremendously by being rich in something called cycles, or almost exact repetitions of patterns. Light is cycle. Electrons going around atoms are cycles. Orbits of planets and bodies are cycles. Vibrations of matter, including the gentle swaying of a pendulum in a grandfather clock, are cycles.
Are cycles simple, then? No, emphatically not! Cycles are walks on the edge of a razor, with chaos on one side and locked-down, frozen simplicity on the other. Planets have cyclic orbits, but add too many bodies or too much time and their simple cycles self-destruct into some form of chaos. But if you go the other way and lock cycles into such extreme sameness that there is no measurable change of any kind from one tick to the next, you achieve not a clock, but perfect oscillator that has no more sense of time than does the world of chaos.
It is only that careful balance of recognizable but slightly different cycles -- that is, of repeated patterns that an intelligence can look at and say "that is still the same light, or that is still the same pendulum, despite the slight changes in position or energy or momentum" -- that makes measurable time possible, and through that allows intelligence -- memory plus meaningful, simulation-like operations that somehow mimic and predict the external world -- to perceive "time." In special relativity this cyclic concept of time is typically represented by the concept of "proper time" $\tau$, which is time as measured by an actual clock.
To achieve length-like time only requires one more comparatively simple step, but that last step also runs the greatest risk of deceiving us. We take our model and out counts of almost-identical cycles, and say "this is like a line, this is like a length. I will represent the progression of cycles as a length, using this distance X I have borrowed from the world I can perceive right now. I will call this axis "time" or $t$, and I will postulate that it exists in addition to the axes of length that I can perceive directly.
It's a very good postulate, and special relativity in particular immediately provides us with some non-trivial substantiation of it by showing us experiments where the easiest way to model the results of velocities near the speed of light as cases where the "time" axis $t$ of the speeding object has been bent and rotated into one of the observer's XYZ axes. But even there, beware! The actual events that get measured are again in terms of cycles -- the cycles or $\tau$ time of the observed object appears (to the observer) to be slowed down. That slowing down can again be mapped into a length-like concept of time, but the mapping still exists. Even there, time as a length-like measurement is indirect in a fashion that should be recognized as part of the process, if you want a more complete picture.
The bottom line of all of this is that if you want to think clearly about complex or advanced concepts of time, don't forget poor old cyclic-only, grandfatherly $\tau$ time as the starting point for all time concepts. It is a deceptively complicated concept, one that says for example that classical physics is just as much "observer dependent" as quantum physics. Why? Because every time you use $t$ in an equation of classical physics, you have implied cycles, and wispy, ephemeral memories of cycles, and remarkable sentient operations that use those misty memories to understand and predict what will happen next, then reason on them.
When you say $t$, you imply $\tau$, and when you imply $\tau$... you imply us.
You have asked more than one question. In your last paragraph, I believe your main question is this:
... if we reverse the roles of time and space here, and instead give information about a single point of space for all of time, it seems we cannot predict spatially. Are there equations in physics that can be considered to predict across space (for a given time)
A space-like slice (subspace) of spacetime stores information, whereas a time-like slice (a worldline) does not. More to the point, the very way a "particle" is defined is an attempt to trim away as much variable information as possible, so that the continuity of conserved quantities such as mass, charge, and spin is emphasized.
In classical physics this focus on reductionism across the time axis leaves you with not much more than the history of how the particle was "bumped," billiard-ball style, as it moves across its path across time. Those deflections provide a small amount of information about the universe as a whole, but the total information encoded is quite trivial compared to that contained in any space-like slice, and is certainly never enough to reconstruct the universe as a whole. The amount of information contained in a single particle path is also highly variable. It approaches zero in the case of a particle that simply sits in a very dark corner of intergalactic space and never interacts with much of anything, so in that case you are left in the dark pretty much about everything else going on in space.
Incidentally, you may wonder why in defining time slices I have focused on the worldlines of individual particles, instead say of a single fixed point in space from the start to the end of the universe. You could use the latter approach, and it gives the same result, since for example a single dark spot in deep intergalactic space has even less information about the rest of the universe than a very bored particle sitting there. However, I don't use that definition because "where" that specific point in space really is quickly becomes entangled with the question of "where it it relative to some set of particle worldlines." Since that is the only way to make such an assertion meaningful, it's easier and more honest simply starting with the particle worldlines.
You also noted in your second paragraph:
... Dimensionality (the fact that there are three spatial dimensions but only one temporal) surely cannot be sufficient ...
Yes. In fact, if you look at what I just wrote, it applies just as well to a 1-to-1 ratio of space axes to time axes as it does to a 3-to-1 ratio. Time is simply the axis of quantity (e.g. mass) conservation, while space is the axis on which the relationships (information) that capture the variable relative configurations of those conserved quantities is expressed.
So what does any of this have to do with my earlier answer about time operating first as cyclic rather than length-like? Quite a bit, actually. The cycles are just repeating patterns of relationships between the conserved, particle-like conserved quantities. So, the conserved-over-time mass of a planet orbits the similarly conserved mass of the sun, and from that detectably similar pattern we define as the year.
It's not that you cannot have change without cycles. It's just that without the concept of some patterns "repeating," you cannot create a truly metric concept of time that like space includes definite lengths and distances. That makes the time version of "distance" rather odd, and a lot more complicated than the space-like version.