Some of this may be just choice of terminology. However, I think there is an aspect of nonrelativistic QM that Matterwave's comment does not address.
If we were talking about classical mechanics, then I would have no problem with viewing the entire spacetime as a single manifold, R^4, with one time and three spatial coordinates. However, in QM (we're talking nonrelativistic QM now), we can't do that because the physics is no longer deterministic: there is not a single unique future for a given 3D spatial "slice" in the spacetime. So you can't just view the spacetime as a single R^4 manifold, because the physics doesn't tell you *which* R^4 manifold it will be; it only gives you a probability distribution over different possible R^4 manifolds, given a set of initial data on a 3D spatial slice. In other words, in plain English, you can't predict the results of quantum experiments definitely in advance; you can only assign probabilities. So your mathematical framework has to take that into account.
In standard nonrelativistic QM, the way this is handled is to describe states of the system we are interested in by labeling each state with its own unique set of "coordinates" (such as x, y, z, if the "system" is just a single particle moving in 3D space and not interacting with anything elsebut there can be lots of other kinds of systems requiring different kinds of coordinates). Then we express the various probabilities for one state changing to other possible states by bringing in this parameter called "time", which is fundamentally distinct from the coordinates.
Once you try to include relativity, the single time parameter no longer works, as I said in that old thread, because there is no single universal time; the "rate of time flow" varies for systems in relative motion. Dealing with *that* ends up leading to quantum field theory, which changes the whole framework again.
