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The Etymonline entry for "soviet" states:

1917, from Russian sovet "governing council," literally "council"…

Likely I am not the only one to wonder about the reason behind the apparent difference between soviet and sovet, but all my attempts to find a proven explanation failed.

There is a dubious claim in Russian Wikipedia that the issue has something to do with old pronunciation of совѣтъ, and that's the point from which I started my research. Actually according to a monograph (pdf in Russian) by Russian phonetist Professor Leonid Kasatkin, yat used to be pronounced as an -ie- diphthong in some positions in old Moscow dialect up to mid-18th century, and Kasatkin himself even attested that pronunciation in Leka dialect as recently as 1992.

However, when this feature of the dialect was discovered in 1912, it came as a surprise, so I'm sure the word in question wasn't pronounced with a diphthong in the cities where Western reporters were working in the early 20th century.

Indeed, I managed to find a number of attestations where the word was transcribed in what I would call a regular way:

• 
  1885: >the Council is called Gosudarstvenny sovet
  


• 
  1892: >It is the State Council— Gossudarstvenny Sovet—a purely consultative body, nominated entirely by the Tzar


• 
  1895: >Tzar contemplates appointing a High Council (Verhovni Sovet)


• 
  1896: >the so-called Council of State (Gossudarstvenny Sovet)


• 
  1909: >Council of Curators (popechitelnyi sovet)… Council of Guardians (Opekunskii Sovet)


• 
  1915: >It was in February 1914 that the question of temperance reform was raised in the Gosudarstvenny Sovet, the Russian House of Lords

and even

• 
  1919: >they proclaimed themselves the Workmen's and Soldiers' Council—the Sovet.

Nowhere in the Google Books corpus this word (nor soviet, which is not attested prior to 1917 according to the OED, as I'm told) is used to denote councils of the 1905 Revolution at the time, despite what is implied in a 2013 answer on this website. What I could find in the sources contemporary to the events are mere translations:

• 
  1913: >Council of Labour Delegates at the end of 1905


• 
  1913: >Such an organisation was the Council of Workmen Deputies. Its story is a story of fifty days. It was constituted on October 26, 1905.


• 
  1914: >There had already been formed among the troops a "Council of Soldiers' Deputies"


• 
  1918: >From the end of 1905 to March, 1917, the Petrograd Council of Workers' Deputies lived underground.

I have a weakly substantiated hypothesis which I am laying out below in the answers (so as not to make this question too long) but I hope the users on this website might have access to better text corpora than Google Books (I wasn't able to check any newspaper articles from the time, for example), and thus may propose better hypotheses.

The orthographic form of the English word may have not actually been borrowed from Russian, but from French, where it is used with the meaning 'council in Slavic countries' since 1840. Besides the attestation mentioned in TNF, I found seven more at 1843, 1844, 1856, 1859, 1865, 1869, 1895 and 1901.

All these books without an exception refer not to a concept from Russia, but rather lead us to Balkans, where a revolutionary government called Praviteljstvujušči sovjet serbski (lit. 'Serbian Ruling Council') was organized by local rebels during the First Serbian Uprising. Other sovjeti were organized later during the 19th century.

The term borrowed into French is not indigenous but has the same provenance as Russian совет: namely, Church Slavonic совѣтъ, from Old Church Slavonic съвѣтъ 'council' (attested in later copies of a text written as early as 9th century). Compare with inherited doublets Ijekavian savjet and especially Ekavian savet, which displaced the Slavonicism in the 1869 Constitution of Serbia.

As far as attestation of 'Soviet' goes, OED (paywalled) abdicates by attesting the noun sense A1a from 1917, and annotating the sense to acknowledge earlier uses (bolding mine):

In the U.S.S.R.: one of a number of elected councils which operated at all levels of government, having legislative and executive functions.
_{The term was also applied to various revolutionary councils set up prior to the establishment of socialist rule in 1917.}

The OED etymology given is "< Russian sovét council"; observe the diacritic, which represents OED's "more 'scholarly'" (see the third paragraph of the "Headwords and the selection of entries" section in the "Preface to the Third Edition of the OED") transliterative romanization of the vowel ye ("Name of letter") in the cyrillic original.

A newspaper search for earlier attestation of 'Soviet' did uncover this from a
"PROCLAMATION" attributed to "Michael Obrenowitsch, Prince of the Servians" in The Times (London), 01 May 1841 (paywalled, bolding mine):

The political events which troubled the tranquility of our country in the past year occasioned several members of the Soviet to lay down their appointments, partly because the people had risen against them, and partly of their own accord.

(The history of this proclamation is, of course, interesting in its own right, but omitted here.)

Serbian, while very similar to Russian, was not identical, and the value of the transliterated use of "Soviet" in The Times amounts only to confirmation that the term 'Soviet' was expected, in context, to signify a type of Slavic "council, board" to an educated, English-speaking audience as early as sometime prior to May, 1841.

Additionally, a search in the Google Books corpus (via HathiTrust) turned up this phonetic transliteration (I assume) of the Russian word 'совет' in the 1833 Ivan Vejeeghen; or, Life in Russia, by Thaddeus Bulgarin:

...who have lately come of age and borrowed three hundred thousand roubles in the Opekoonsky Sovyet?*

....

* The name of the bank belonging to the Foundling-hospital.

Later, in the first decade of the 20th century, I found uses of 'Soviet' in 09 May 1905 The Indianapolis News, Indianapolis, Indiana (pages 4 and 6, emphasis mine). The first is a syndicated article distributed by the Associated Press; the second is a semi-humorous 'blurb':

NEW COUNCIL FOR RUSSIA IS FORMING

Character of the Branch of the Government Provided by Czar's Rescript is Defined

"ZEMSKY SOVIET" IS PLANNED

Membership Will Be 500 Men to Serve as Intermediary Between People and Ruler.

ST. PETERSBURG, May 9, 3:07 a.m. — A report defining the form and character of the new branch of the Government to be created in accordance with the imperial rescript of March 3 is current in official circles here, according to which the fruits of the Bouligin Commission have taken the form of a resolution for the creation of a "Zemsky Soviet" or council of Zemstvos, elected indirectly through the Zemstvos, to serve as a connecting link between the Emperor and people.

....

Though such a Zemsky Soviet will fail to satisfy the Liberals, who are calling for a full plunge into constitutionalism, universal suffrary, a responsible ministry, etc., ....

---

[Page 6:]
The latest dispatches announce that it is a Zemsky Soviet which is to be created in Russia. Presumably it was discovered that a Zemsky Sober was impracticable.

From an editorial titled "Russia's Awakening* in the 14 May 1905 The Indianapolis News:

....

 Now comes also the entering wedge for constitutional government. The Bouligin Commission appointed some months ago by the Czar has reported a scheme for a popular assembly of an advisory character, to be known as the Zemsky Soviet.

Another syndicated article, datelined St. Petersburg, 18 May 1905, was picked up by two newspapers in the Newspapers Extra corpus:

WILL HE MAKE GOOD.

It is up to the Czar to Keep His Promise.

...All St. Petersburg and Russia in general are waiting today to see if the czar will make good his promise of last March to form a new branch of the government to serve as a connecting link between the emperor and the people. The new branch will be called the "Zemsky Soviet" and today has been set as the date of the proclamation putting it into effect.

These were followed by more widespread appearances in May and late October, 1905, in short syndicated articles documenting the creation of the Zemsky Soviet. One of the May articles, in the New-York Tribune, seemed to have been editorially altered to use the 'Sovyet' variation of 'Soviet'.

Of the October articles, another syndicated (AP) article appearing in the 26 Oct 1905 Leader-Telegram, Eau Claire, Wisconsin and other newspapers, describes 'soviet' as the "old name":

....

 It is understood an edict formally establishing a ministerial cabinet which will bear the old name of "soviet," or council, will be published....

Substantially the same article appeared on 26 Oct 1905 in The Billings Gazette; that article had been locally edited to delete "old" before "name".

At about the same time (Sep-Nov 1905), articles in the popular press of both the UK and the US made mention of the Soviet Siezd (Council of Associations of Naptha Producers; Siezd is also spelled Sieza in these reports).

The tendency of these (and, presumably, other) mentions of 'Soviet' in the popular press was to firmly entrench the term as the English for a "Slavic council or governing body". At the same time, the occasional use of the orthographic form 'Sovyet' suggests the word was at least perceived by English speakers as having been adopted from Russian or another Slavic language.

Altogether, early uses, including those I managed to uncover in the first half of the 19th century and the first decade of the 20th, along with the general sense that the more usual phonetic representation in roman characters was -ie-, may have led to -ie- "winning out" over the simpler -e- for transliteration of the cyrillic сове́т. As commenter @sumelic made clear,

Russian speakers don't hear "ye" as in "yet" in this word, although English speakers do; to a Russian speaker, it's about the "v" being "soft" or "hard".