Plants are remarkably "plastic". If you do any gardening you will soon see that plants can be propagated from minute quantities of parent material. A common weed in UK gardens is ground elder (Aegopodium podagraria) which I first met as a child while weeding for my parents. The smallest part of rhizome left in the soil would soon grow into more plants. Japanese knotweed (Fallopia japonica) is reputed to grow vegetatively from less than one gramme of plant material (don't try it at home because you'll never get rid of it). The point of explaining this is that some plants can't really be considered dead while there is life lingering in one cell. If we were like plants we could sever a finger, grow it back AND, with a bit of TLC, grow a whole new person from the finger.

In short: no, they don't die suddenly like humans.

It is hard to answer that without butting a large number of buts. As Pete said, if we ignore what happens if you do a plant violence (like burning or squashing it) they generally die slowly and leaf by leaf or branch by branch, but really, much the same is true for humans too. We do have certain organs (such as heart or brain) without which we die quickly, but more often we find some of our organs failing slowly so that we die like a rotting oak or sick potato. (Ask the Irish about Phytophthora infestans, which can kill a plant on a matter of days.)

But we don’t just die instantly. Just as with plants, many of our cells will stay alive for varying lengths of time, usually until the supply of ATP is exhausted. The reason it seems otherwise is that we define death in humans as either the heart or the brain stopping their normal functions. SInce a plant has no equivalent organs, we are not as aware of what is happening to them.

One of my hobbies is to grow and collect somewhat unusual plants such as palms and succulents.  I live in an area of SW France where the summers are very hot and dry and where the record minimum temperature is -18° C and the record maximum is 40° C.  If I do not protect plants having little cold tolerance, just one night's hard frost will kill them.  Equally, accidentally forgetting to water one for a few days with the temperature hovering around 40° C produces  the same result.  There can be little doubt that a plant frozen overnight with all its cells smashed is dead.  Some plants subject to such treatment, however, may appear to be alive for some days afterwards until they begin to discolour and wilt that is.  As for the hot summers, trees kept in pots are particularly susceptible.  Just a few days without water and they are finished.  Realising one's mistake and watering makes no difference, just a few days and they are clearly dead as a dodo.