The question itself is enormous, and therefore what follows can but skim the surface of it; however the lines of research that are indicated can be emulated to make further discoveries.
The three words are similar in that they all serve to formulate an appreciation about a state of things of relative importance, that appreciation being to the effect of caracterising that importance as well above the average point. Where they differ essentially is in the domain of concepts they can be applied to. Two of the words are similar in this respect; they are "immense" and "énorme".
"Énorme" comes directly from latin (enormis) and is found in two domains of signification, the quantitative and the qualitative (ref).
a/ In the quantitative domain it does not mean "deprived of norm" as the sense of the latin prefix (e-,ex-) could entail but "far from the norm" (another of the senses ot this prefix); the definition makes precise that this removal from the norm is in the upwards direction: "above the average", "much bigger than what is usual"; it is used to modify the following elements;
- nouns of material things: arbre, ville, nuage,
- abstract things that imply a collective notion: population, foule, troupeau, audience, …
- abstract things that can be measured: masse, hauteur, distance,…
b/ In the domain of the qualitative several notions are piled up on this form. One particularity of its use is that often it is placed after the noun rather than before, but there is no rule, as can be seen from the ngrams below;.
- unusual, excessive:
- great importance: un rire énorme (ngram)
- grievous: une erreur énorme (ngram)
- abusive, tasteless: une plaisanterie énorme (ngram)
- greater on the basis of merit or quality: une œuvre énorme (ngram)
"Immense" means basically
1/ a/ "that which is limitless" (Dieu est immense)
b/ By extension of the concept the word has come to mean "of very great extent in its dimensions", "vast".
2/ There are also a few figurative senses.
"Immense" is used rather for expanses of land and water, areas, …
"La mer est immense."is more expressive than "La mer est énorme." (ngram)
étendue immense, étendue énorme (ngram)
immense étendue, énorme étendue (ngram)
malheur immense, malheur énorme (ngram) (For instance "dans leur malheur énorme" is almost never said whereas "dans leur malheur immense" is acceptable.)
In order to describe massive objects both adjectives tend to be equally usable but "énorme can be preferred.
montagne énorme, montage immense — énorme montagne, immense montagne (ngram, ngram)
une énorme boite, une immense boite (ngram); apparently, "boite énorme" n'est pas utilisé (ngram).
"intense" is used to refer to a relatively great importance of a phenomena that presents homogeneity and that can occur at any level of a continuous distribution of force or intensity that is naturally found for these phenomena: heat, current (fluid, electricity, air, …), light, …
We see from this, then, that "intense" is also used to connote a norm that is above the average, but in a particular domain; it can be considered as a specialised form of the other two words; as most often in this type of relation the specialised word tends to be restricted to its own domain of specialisation while the general word can eventually infringe upon that domain.
chaleur intense, chaleur énorme (ngram) (There is here the problem of knowing what's meant by "chaleur", whether it's temperature or quantity of heat.)
température intense, température énorme (ngram)
intense température (ngram) (Strangely enough, it can be seen here that the consecrated term is not at all favoured as a preposed adjective in the syntagm.)
immense courant, intense courant, énorme courant (ngram)
lumière intense, lumière énorme, lumière immense (ngram) (Here the theory finds confirmation.)
recherche intense, recherche énorme, recherche immense (ngram) (Here again, confirmation)
intense activité, énorme activité, immense activité () (Here again, confirmation)
étude intense, étude énorme, étude immense (ngram) (Here also, confirmation)
intense malheur, malheur intense (It appears that misfortune is not considered to be a gradable phenomenon.)