If giving a reasonable answer requires imposing some restrictions
(only verbs and only in classical Latin, for example), feel free to do
so. Also illuminating examples are good even without an overall
theory. I am looking for some intuition into this thing, not any
Here is my intuition.
Intensifiers can derive their meanings from different underlying metaphors that, even when no longer consciously felt by speakers, can still affect their uses and nuances. Compare "she is very/really/terribly beautiful." I don't think these words clearly differ in degree, but do carry different nuances. The origin of "very" is most removed from my conscious use and has only the meaning "to high degree" and no connection with its origin in words meaning "true," even though "very" can still be used in archaic expressions like "she is the very soul of beauty." The word "really" conveys that the degree contradicts possible expectations and so is "real" despite what one would think. The word "terribly" invokes strong emotion and so suggests that the degree of beauty is such as to arouse emotions. I see the difference between per- and dē- as stemming from similar differences.
I think that verbs with Latin prefixes stand in the language in roughly the same position that phrasal verbs do in English, and therefore I start with my English intuition about phrasal verbs. These verbs seem to consist of metaphors originally grounded in physical relationships, but often suggesting more.
The meaning of a particular phrasal verb expression often cannot be expanded into a simple phrase that illustrates its origin, and even if that were possible, such an origin is not necessarily transparent to a native speaker. For example, I cannot think of plausible expressions that would explain the origin for the expression "to put up with"; nevertheless, the individuals parts of the expression do seem associated with other meanings of these words.
A good example of such verb phrases in English is the pair "tire" and "tire out." They don' really differ in degree so much as differ in focus. The plain verb merely states the nature of the activity, while the phrasal verb adds the concept of the end state. If you say: "I tired him," you do not clearly refer to an end state. If you say: "I tired him out," it is clear that he was left tired.
In Latin, per seems to have the base meaning of "through." As a prefix, it often denotes completeness, presumably with the underlying metaphor that when you go through something you have reached the end and traversed its complete dimensions. I associate it with meanings like "through and through," "to the end," and "thoroughly." With this basic understanding, I understand a verb like pervincō to mean something like "defeat the full extent of" or sometimes "to get through a situation and end up with victory."
With dēvincō, Wiktionary has "defeat decisively," perhaps emphasizing the changed state of the object or the conclusiveness of the defeat. Also, adding dē can slightly alter the semantics, as in dēvicta bella (Verg. A. 10, 370). Wars are not themselves defeated or conquered, but they can conclude in defeat or conquest. The focus on the end state can better license using "war" as an object of the verb, rather than an enemy.
The prefix per can also introduce other nuances such as in pereō and perdō. In these words, I think the nuance is a negative one stemming from "to go clean through something without effect." With this nuance, pereō ("to perish or be ruined") would mean "to go all the way to the end of life or wellbeing" or "to go so far that you are through." The word perdō ("to lose") would mean to place something so that is through or beyond usefulness. In perdōnō, perhaps calqued from a Germanic expression ancestor to "forgive," we might have the nuance "give grace through it all."
With dē, we have a word meaning "from," but perhaps emphasizing the source. From this meaning, we get the further meaning of "down from." I associate it with English words like "away" and "down." In compounds, it often suggests that something is no longer in the position it should be, sometimes in a lower place. That is how I understand dēplōrō, as not just weeping over something, but emphasizing the bad circumstances that something has reached.
For populor, we have a word meaning "to pillage or lay waste." If we add "per," we empasize the completeness and that nothing is excluded from the effects, such as applied to "all places" (“omnia loca,” Livy 34, 28) a large area like "Italy (“Italiam,” id. 22, 3; Tac. A. 14, 26) or "a field" (“perpopulato agro,” id. 22, 9, 2.). These places are thoroughly pillaged.
If we add "dē" to the root popula-, we seem to stress the changed status caused by the act. With this understanding, a word like populatiō ("plundering") would stress the action and dēpopulatio would stress more the effects ("ravages"). If we take this over to the verb, dēpopulo(r) would be more like "to ravage and leave devasted."
Taken another way per would stress the horizontal range of the pillaging through space or time and dē would stress it vertically.
With some words, only one prefix makes sense. For instance, tristis does not emphasize a change in state and so does not readily combine with dē, but makes good sense with per as pertristis to mean "sad in all respects" or "thoroughly sad."
With dēfetīscor (grow tired), we have a word stressing an altered state, with no related form *perfetīscor. The latter would be a good form to express "tired all over," but apparently did not exist.
In dētrītus, on the other hand, we have a new state in "something worn away/down." But we have pertrītus in the meaning "to rub, bruise, or pound to pieces" where we stress that the whole object is affected.
With plōrō, we have a verb that can take an object to mean something like "wail over." My guess is that it creates a quite physical image in the mind of an action simultaneous with the stimulus, even if used metaphorically to mean "complain." With dēplōrō, I think we have a verb that points more to the cause or source of the action and is more definitely subsequent to the stimulus and slightly more removed from it. It probably also conjures up the altered state of the stimulus more than just the response to it.
In short, I think per and de originally conjured up different metaphors. In time, such things can be emptied of all nuance and just leave the meaning of intensification, as happened to "very" in English which carries no sense of "in truth" as might be expected from its origin. In the clasical Latin we see, I don't think that stage had been reached yet.