If time in physics is static and just one of the dimensions, what exactly makes it different from space in physical models?

The key differences is that one can predict across time, but not across space. From a photo of a room one can easily see what happens in one hour. Most things will be at the same place, the shadows from a sun will slightly move, a cat sleeping in a chair will not be there. But that is it.

Now suppose one is gets a one-hour video of a wall in the room and the question is to describe the whole room at the end of that hour. The answer is that one cannot. As with the story of Plato’s Cave at best one can speculate from a shadows on the wall. But the speculation will be not much better than random guesses.

Physical equations reflects that. Speaking mathematically the equations are hyperbolic or parabolic in nature, but this is just precise expression in the language of math of the notion of predictability across time. Another way to say this is that each particle follows its worldline and one in principle can calculate the worldline for each particle from initial conditions both into future and past.

But this predictability is lost in General Relativity at singularities. It follows from the equations that in the center of black holes or at the Big Bang it is not possible to predict. One cannot draw the worldline through singularity. There literally anything goes.

Which explains why the singularities of General Relativity are considered bad. Our experience tells us that predictability across time is always there. Thus those singularities must be an artifact of the model and should be corrected with a better theory.

Yet General Relativity survived more and more stringent tests and now we can even detect gravitational waves from the merge of black holes. So maybe past and future are not always predictable.