Learn English – Difference between *guile* and *beguile*

You can think of grudge as a hypernym of resentment.
A grudge need not be the result of (perceived) unfairness; it can just result from (violation of) the holder's vested interests irrespective of the fairness involved. Also, a grudge is always directed towards people (I haven't seen otherwise).
A resentment on the other hand (as you said) results from a perceived unfairness. A resentment, unlike a grudge, can be directed towards concepts (like policies and laws) in addition to people.

Definitions from M-W to conclude:

grudge noun

: a strong feeling of anger toward someone that lasts for a long time

He has nursed a grudge against his former boss for years.

resentment noun

: a feeling of anger or displeasure about someone or something unfair

He expressed his resentment of the new policies.

Note from the examples:
The grudge does not necessarily mean that the boss was unfair.
The resentment is directed at policies.

It is not uncommon for technical subjects to assign specific senses to words which would be considered close synonyms in English. For example, a thesaurus would tell you that force, power, and energy are all synonyms; while they all have distinct meanings in Physics.

In your case, denote is the word typically used in math/science to describe the symbol associated with something, (see sense 3); thus distinguishing it from represent, which is employed for diagrams and such.

In thinking that the words should be interchangeable, you might be considered guilty of genetic fallacy.

The genetic fallacy (also known as the fallacy of origins or fallacy of virtue) is a fallacy of irrelevance where a conclusion is suggested based solely on someone's or something's history, origin, or source rather than its current meaning or context.

_{Wikipedia (emphasis mine)}