Wonderful question! The only material difference between a living and dead cell is that the living one works - all chemical reactions are regulated, sustained and limited, synergetically contributing to the maintenance and continuation of the existence of the functioning cell. In a dead cell, chemical reactions are runaway and tend to the destruction of the cell as a whole. That does beg the question of what is a 'sick' or 'cancerous' cell, of course.

"life", "alive" is notoriously difficult to define.  words like "replicators" (dawkins), or "growth" have been used.  there are exceptions to every definition.  "organismizations" such as "prions", "crystal" (growth + replication), "gaia", "lateral gene transfer", "eukaryote", "species hybridization" are all key words that can make you wonder more!  isn't nature wonder-ful?

I believe teachers tell that story in a wrong way. What does it mean that there is no chemical difference? If that means that there is the same amount of carbon, and nitrogen, and oxygen and so on well that could be the truth for a short period of time, but if that means that there are the same amounts of all different molecules, that they are arranged in the same way, that they all have the same shape, that they have the same electrical charge, that they interact in the same way they interacted in the living cell - well, that cant be right! Take prions - there is no chemical difference between a normal protein and a prion, all aminoacids are arranged in the same order jet prion has an unusual way of folding and cant function properly in the cell  and you end up with CJD. Smashed car is not chemically different from a new sparkling car that just arrived from the factory but it is "dead", it doesn't function properly and you end up walking instead of driving.   I think that you cannot just assume the cell is alive and well one moment, and dead the next moment with nothing in between. There is a dying period no matter how fast. Period when violent reactions take place, when dynamic reactions in the cell are altered and the cell is "struggling" until it looses the battle and entropy wins, or until it reaches the normal equilibrium of events and lives on. I like to think that cells are not really, really dead if they can still maintain their membranes intact. Perhaps dying, but not jet dead. Like some doomed character from classic tragedies...

A dead anything is the same as it was when it was living, just moments ago, with respect to its constituent particles, like cells or enzymes or chemicals. However, they may all be jumbled up (like when one gets shot) and thus can't function properly, or they may have been stopped (as in freezing) or thrown around violently (as in boiling).

So, the difference is not in the matter (the living thing is made of the same stuff as the dead thing), but where it is and how it is moving.

Good question. I'm not an expert but I'd say it's analogous to a car running down.

Different systems start to malfunction over time but the car still runs. Eventually it reaches a point where perhaps it starts to cough, it doesn't go over a certain speed, some gears can't be used. At some point it stops being able to move, even though the motor might still run. At what point does the car "die"? There's no definitive cut-off point; the car runs-down over time.

Similarly, perhaps, the many functions of a cell stop working or slow down. It doesn't just die at a given moment, it just runs down over time.

Which also implies that there may be many ways that a cell can "die", just as there are many ways that a car can run-down. There are lots of tiny differences between live and dead cells, and these differences may be different from cell to cell.

As I say, I'm not an expert, I have an undergraduate degree in psychophysiology which involved a fair amount of study of cells, mainly neurons. But it seems feasible.