What is the difference between parataxis and asyndeton?

Online, I have found many different definitions and comparisons of parataxis and asyndeton. In the dictionary (New Oxford American) they are defined as follows:

Parataxis: the placing of clauses or phrases one after another, without words to indicate coordination or subordination

Asyndeton: the omission or absence of a conjunction between parts of a sentence

I do not see a difference in meaning between these definitions. Curiously, some webpages claim parataxis may include conjunctions.

LiteraryDevices.net describes the devices like this,

Parataxis is derived from a Greek word that means “to place side by side.” It can be defined as a rhetorical term in which phrases and clauses are placed one after another independently, without coordinating or subordinating them through the use of conjunctions. It is also called “additive style.” Parataxis is sometimes used as asyndeton, in which the phrases and clauses are coordinated without conjunctions.

Again, "without coordinating or subordinating them through the use of conjunctions" seems to be equivalent to "coordinated without conjunctions." If, perhaps, there is no coordination regardless of conjunctions in parataxis, how is it possible to distinguish?

What makes "I came, I saw, I conquered" as three independent clauses arranged side by side distinct from reading the statement as "I came, I saw, and I conquered," only with the conjunction removed?

Maybe parataxis can apply to separate sentences as independent clauses, e.g. "I came. I saw. I conquered," while asyndeton is limited to the parts of a single sentence?

There is, I fear, a problem with these terms and there relationship. But, on the evidence I have, they are different. First, there is a clear meaning for asyndeton. From the Greek ασυνδετον, meaning non-co-ordination. It is a stylistic device of rhetoric. Observers of proceedings of the British parliament will hear a great deal (some might say too much) of it. It involves the piling up of grammatically parallel words, phrases or even clauses.

Julius Caesar famously boasted in three words:

Veni vidi vici = I came I saw I won (Romans did not punctuate at this time)

The OEG gives the following example of asyndeton

Agricultural crops would be scorched, and yields would fall; marine plankton would be affected; human health would suffer (there would be more cataracts, more problems arising from damage to people’s body immune systems.

You can see that the first two clauses, linked by a conjunction, used, as often. for a consequential connection. It is syndetic. What follows is asyndetic. The rest of the sentence and the sentence as a whole are asyndetic.

There will be an agricultural disaster; there will be a marine disaster; there will be a human health disaster, consisting of three problems connected by anaphora (repetition of “more”), piling up the bad outcomes. But what is in the bracket is asyndeton as well as anaphora.

Greenbaum explains parataxis in contrast with hypotaxis as follows:

Parataxis is a relationship between two or more units that are of equal grammatical status, as in co-ordination (books and magazines), whereas hypotaxis is a relationship two units, one of which is dependent on the other, as in modification (the relationship between the relative *that-*clause in books that I have read).

So the answer to your question is that they are different, even though they often occur in the same sentences. parataxis (and, less often hypotaxis) is about syntactic relationships, where what might be called syndesis is a matter of the presence or absence of a conjunction, modalising or otherwise.