Let's discuss the development of the symbol. It might not surprise you to see that there's a connection to Hebrew here as well...
We begin with the Egyptian word ntr (I've also seen it spelled netjeri / netjry /ntrj / ntry). The word originally meant "divine" or "pure". It referred to a salt called sodium carbonate, also known as "washing soda" or "soda ash" (and sometimes simply "soda"). It was found in the area of Wadi El Natrun (apparently the wadi was named after the salt, not the other way around), in salt lakes. It was used both for soap as well as in the process of mummification.
From Egyptian the word was borrowed into Semitic languages - Akkadian nit(i)ru, Aramaic nithra, and Hebrew neter נתר. This word appears twice in the Bible - Yirimiyahu 2:22, where it talks about its use in cleaning and Mishlei 25:20, where it describes the effect of mixing it with vinegar (remember those volcano experiments from grade school where you mixed baking soda - sodium bicarbonate - and vinegar? Same thing here.)
However, in Israel, outside of Egypt, neter didn't only refer to soda (sodium carbonate) but primarily to potash (potassium carbonate), which was derived from plant ashes. (Note that we've previously discussed the Semitic origin of the symbol K for potassium, and its connection to plant ashes.) By Talmudic times (Shabbat 90a), they distinguished between the two types: Alexandrian neter, which came from the Egyptian salt lakes, and Antipatrisian neter, which came from the town of Antipatris - made from the ashes of plants.
The Greeks also had a term for this kind of soda - nitron (sometimes spelled litron). There's some debate as to whether the term was directly borrowed from Egyptian or perhaps entered Greek via Hebrew / Phoneician. From Greek the word entered Latin as nitrum and Arabic as natron.
If we continue following the word in the European languages, we see that from here it splits into two different paths. The Arabic natron enters as is into Spanish and then French, where it meant both potash and soda. The word natron is still used in English (where it appeared first in 1684), but mostly to refer to the older use of the salt.
In the 18th century, chemists began the process of identifying the various chemical elements. As discussed here, there was a split between Germanic speaking countries and the English and French speaking countries. Sir Humphry Davy was able to isolate the element sodium. As it came from "soda" - sodium was the name he chose, and this is the name used in French and English speaking countries. However, the Germanic speaking countries follow the chemists Gilbert and Klaproth who first called the element natronium (the suffix -ium means "metal", i.e. this is the metal component of soda), and then later Berzelius who came up natrium. And as we saw with Potassium and K, while the name of the element is sodium in English, the symbol of Berzelius was adopted - Na.
In Hebrew the word for sodium is natran נתרן. I haven't seen its coinage discussed extensively, but I imagine that on the one hand the influence of German science is felt here, as well as a desire to connect it to the Biblical word neter.
We've now followed the path of the word natron. However, we find another word that derived from the Greek nitron and Latin nitrum in European languages - nitre. The word is first found in French, and from there to English, where it is spelled nitre in the UK and niter in the US. Nitre does not refer to sodium or potassium carbonate, but potassium (or sodium) nitrate - better known as saltpeter (US) or saltpetre (UK). This substance isn't used as a detergent, so the translations of the Biblical neter as nitre are incorrect. Nitre was used as a fertilizer and as an explosive (for an interesting description about the connection between fertilizers and explosives, read Pollan's The Omnivore's Dilemma.)
Again in the 18th century, Rutheford discovered the chemical nitrogen, but the name was given by the French chemist Chaptal because it produces ("generates") nitre. So we actually have two elements - nitrogen and sodium, that derive from the same word.
The only question that still remains is why did nitre come to mean potassium nitrate instead of sodium carbonate? I've seen a few theories. This book (The World's Greatest Fix: A History of Nitrogen and Agriculture by G. J. Leigh, page 89) tells us that
the alchemists who first described these salts were not then able to distinguish clearly between the two types of salt.It also points out that "nitre" stopped meaning "natron" in English in the middle of the 17th century, around the time that English began using the word natron (which must have helped with the clarity necessary to begin identifying elements like sodium and nitrogen in the following century).
This article indicates that the confusion between nitre (saltpeter) and natron (sodium carbonate) goes back very far:
It seems established that Greek nitron and Latin nitrum were used for both saltpeter and soda, which were not recognized as different substances. Pliny definitely used nitrum for both.
With better chemical knowledge, brought to the West by the Arabs, came also an Arabicized form of nitron, natrun, which existed beside the native Arabic word for soda, qali, meaning a plant ash, or milh qali "ash salt." As now two words were available, and two substances were distinguished, soda was called natron, while nitrum was specialized for saltpeter.
One other theory is presented by this book from 1844 (I don't know how well it has aged):
Their nitrum, however, must have been exceedingly various in its properties. For this incrustation is not always calcareous saltpetre; it is often soda, mixed with more or less calcareous earth ; and sometimes it consists of salts of sulphuric acid.At this point the answer of when the terms diverged is not entirely clear. But I do very much like the quote above: "many wish to have the pleasure, if not of forming new species, at any rate of giving new names". I suppose if it wasn't so, I would have a lot less to write about...
Substances so different ought not indeed to have been all named nitrum ; but before natural history began to be formed into a regular system, mankind in general fell into an error directly contrary to that committed at present. Objects essentially different were comprehended under one name, if they any how corresponded with each other even in things accidental. Whereas at present every variety, however small, obtains a distinct appellation; because many wish to have the pleasure, if not of forming new species, at any rate of giving new names. The elephant and rhinoceros were formerly called oxen ; the sable and ermine were named mice, and the ostrich was distinguished by the appellation of sparrow. In the like manner, calcareous saltpetre and alkali might be called nitrum. The ancients, however, gave to their nitrum some epithets, but they seem to have been used only to denote uncommon varieties.
But were the ancients, under the ambiguous name of nitrum acquainted with our saltpetre? There is certainly reason to think that it became known to them by lixiviating earths impregnated with salts. There are, as already said, not only in India but also in Africa, and particularly in Egypt, earths which, without the addition of ashes or potash, give real saltpetre, like that of the rubbish-hills on the road from new to old Cairo, and like the earth in some parts of Spain. It is a knowledge only of this natural kind of saltpetre, which required no artificial composition, that can be allowed to the ancients, as it does not appear by their writings that they were sufficiently versed in chemistry to prepare the artificial kind used at present.
But even admitting that they had our saltpetre, where and by what means can we be convinced of it ? Is it to be expected that any of the before-mentioned characters or properties of this salt should occur in their writings? They neither made aquafortis nor gunpowder; and they seem scarcely to have had any occasion or opportunity to discover its deflagration and the carbonization thereby effected, or, when observed, to examine and describe it. No other use of our saltpetre which could properly announce this phenomenon has yet been known. How then can it be ascertained that under the term nitrum they sometimes meant our saltpetre? Those inclined to believe too little rather than too much, who cannot be satisfied with mere conjectures or probabilities,but always require full proof, will acknowledge with me, that the first certain accounts of our saltpetre cannot be expected much before the invention of aquafortis and gunpowder. It deserves also to be remarked, that the real saltpetre, as soon as it became known, was named also nitrum ; but, by way of distinction, either sal nitrum, or sal nitri, or sal petrae. The first appellation, from which our ancestors made salniter, was occasioned by an unintelligible passage of Pliny, which I shall afterwards point out. The two other names signify like sal tartari, sal succini, a salt which was not nitrum but obtained from nitrum. Sal-nitri, therefore, or salniter, was that salt which, according to the representation of the ancients, was separated by art from nitrum, yet was essentially different from the nitrum or soda commonly in use. Birin-goccio says expressly, that the artificial nitrum, for the sake of distinction, was named, not nitrum, but sal nitrum.
In the course of time men became acquainted with the purer, more useful, and cheaper mineral alkali which was furnished, under the name of soda, by the Moors and inhabitants of the southern countries, who had learned the method of preparing it. The vegetable alkali also was always more and more manufactured in woody districts, as an article in great request, and sold under the name of potash, cineres clavellati. All knowledge of the impure alkali from the incrustation of walls was then lost ; and as there was no further need of guarding against confusion, it was not longer thought worth while to name saltpetre sal nitri: it was called nitrum and the oldest signification of this word being forgotten, it was admitted without further examination, that the nitrum of the ancients was nothing else than our saltpetre.