One of the best articles I'd read on the subject addresses a few things that you bring up perfectly. It's a bit dated, but it's "Concerning the 'Revival' of the Hebrew Language" by Jack Fellman. As he notes, Hebrew never died, though it was very close to, relatively speaking. Unlike the Akkadian languages with a limited inventory of texts and no native speakers, Hebrew did not die out in the same way. Because the Biblical and Mishnaic forms of Hebrew lived on, and because it was still used as a form of written communication by some in the years following its near (or full) death as a spoken language early after 200 AD, it never fully died out.

Writings persisted in Hebrew, typically taking what Fellman describes as a "romantic" form, but by the middle of the 19th century the writings had become more "realistic", as novels, short stories dealing with contemporary life, and newspapers were published in Hebrew. However, as one can imagine, using Biblical Hebrew for modern times was awkward and unusual, so some words were adopted from other European languages as a stopgap...but this didn't really work all too well.

Then came Eliezer Ben-Yehuda, widely considered the father of modern Hebrew. He wrote an article in the Hebrew periodical Ha-Shahar in 1879, titled "A Serious Question", where he talked about emigration to Palestine as the solution to the Jewish question of where to live as a people. This was not new. However, Ben-Yehuda did introduce the idea of leomut, nationalism, as well as the word itself, in this article. He then talks about using the Hebrew language as a basis for nationalism, as the European model does.

He went to Palestine in 1881, settling in Jerusalem and vowing to only speak in Hebrew with Jews. In 1882, he had a child, and became the first child in nearly 2,000 years to have Hebrew as a first language. Ben Yehuda continued to preach his idea of connecting language to nationalism, and even attempted to compile a dictionary of his own to modernize the language, which was ultimately unfulfilled but which still provided many, many ideas. He created new words based on Semitic roots and stems, and created a Hebrew Language Council in 1890.

Ben Yehuda got lucky. Right after he arrived, so did the First Aliyah, filled with Jews who had left pogroms in Russia and who were eager to take up a new language and test his ideas. They were passively familiar with the language from prayers, but Ben Yehuda wanted their children to speak it too. Still, he pushed teachers (many of whom had never used Hebrew themselves spoken-wise) to teach children how to speak Hebrew in their schools. By the end of WWI, 40% of the country used Hebrew as a first or daily spoken language, and Hebrew became an official language under the British Mandate, alongside Arabic and English. So, while the language was never fully dead, it still required large efforts to push it back into mainstream use and to make it a spoken language that could be used in modernity.

That quest was not easy. There were difficulties because of the ways Hebrew had to be adapted. As Arthur Koestler, a British author and journalist wrote:

I have spent on and off altogether some four years in Palestine and speak Hebrew fairly fluently, but am still incapable of reading a newspaper, to say nothing of books. The majority of new immigrants are in the same position.

The problem, he expanded on in the late 1940s, was the alphabet. It was cumbersome and difficult to use, he claimed, and had to be "latinized". This would mean using the Latin alphabet, also known as romanization. The idea had been pondered at least as early as the 1920s, and one of its biggest proponents was actually Ben Yehuda's son, the child I mentioned earlier! Itamar Ben Avi, as he was called, argued that his father would've wanted the language romanized, though Ben Yehuda had turned down including (in the newspaper he edited) a single page per week in roman lettering. Ben Avi was not the first or the last to try this, but the movement ultimately failed. The difficulty of the Hebrew alphabet was simply tackled head-on. Its role in national identity was simply too strong, even though head of Revisionist Zionism Vladimir Jabotinsky eventually came to back romanization of the alphabet. They failed to convince the majority of Zionists, most of whom were secular Labor Zionists, to support romanization.

The real process of adopting the language didn't just come from Ben Yehuda's modest Hebrew Language Council. A real Hebrew Language Council was founded in 1904 by the Eretz Israel Teachers’ Union, which intended to build a "society of linguists". It was not until 1911 that they really achieved too much, deciding only then amongst themselves that they had the power to invent new words and grammatical forms. Ben Yehuda would recommend the guiding principles that the council would vote on and accept. First, they would try to get Hebrew into all spheres of life. Second they would safeguard its original pronunciations (letter pronunciations), and its "oriental quality". Third, they tasked themselves with making Hebrew flexible enough to discuss any aspect of human thought.

Old words from Hebrew, lesser-known ones, were dredged up and modernized and published. If a word could not be found that corresponded to something, a new word was created, usually trying to use Hebrew roots where possible, with roots from the Aramaic cognate if no Hebrew root could be used, and they would Hebraicize the Aramaic root wherever possible in pronunciation, gender, and spelling. Ben Yehuda decided to rely heavily on other Semitic languages himself, and the Council was not opposed to this at all. As Yehuda described in 1918:

Arabic, in particular, was a kind of source of salvation for me in the linguistic research of our language. First, because it lives at this moment, we are standing on solid ground when explaining the meaning of its words. Second, as I have already proved in one of my papers at one of the Language Council meetings...very ancient grammatical forms of all the Semitic languages have been preserved for us in Arabic; and its very expansive and rich vocabulary is the joint treasury of all the Semitic languages. The deeper I went into Arabic language research, the wider the gates of understanding of the Hebrew language opened before me; the Arabic vocabulary enabled me to discover the authentic explanation of many biblical words...

The HLC did receive some pushback from the Second Aliyah immigrants, most of whom were the first secular Labor Zionists, and who viewed the "old" and "self-proclaimed" Jewish intellectuals on the council with distaste, at first. The question of what language to use exploded in 1913 and 1914, as schools argued over whether to use Hebrew or other languages (like German) for instruction. The HLC played little part in the arguments, though the Teachers' Union did coordinate all the movement in favor of using Hebrew, without talking to the HLC about it.

Still, the HLC never got to prominence like it could have, and the task was mostly left to undertake itself with the roots Ben Yehuda had placed. The HLC did not convene during WWI due to the war itself, despite recognition of a change in the World Zionist Organization's language policy right before the outbreak of the war that centered on "one nation and one language", that language being Hebrew.

I can't speak too much as to the difference, though I do know it's not all that wide in terms of roots, but is commonly characterized as similar to reading Shakespeare vs. modern English. What I can say is that learning Hebrew, while difficult at first, was a process that was refined and altogether common already by WWI, as more and more children were educated in all-Hebrew schools and it became an official language under the British Mandate in the 1920s-on.

Sources:

Attempts at Romanizing the Hebrew Script and Their Failure: Nationalism, Religion and Alphabet Reform in the Yishuv İlker Aytürk Middle Eastern Studies, Vol. 43, No. 4 (Jul., 2007), pp. 625-645

Concerning the 'Revival' of the Hebrew Language Jack Fellman Anthropological Linguistics, Vol. 15, No. 5 (May, 1973), pp. 250-257

Revisiting the language factor in Zionism: The Hebrew Language Council from 1904 to 1914 İlker Aytürk Bulletin of the School of Oriental and African Studies, University of London, Vol. 73, No. 1 (2010), pp. 45-64

I can't speak to most of this, but I am a Biblical Hebrew scholar and may be able to shed some light here about the differences between Biblical Hebrew and Modern Israeli Hebrew.

First, we must understand that the language was never really "dead". There have always been Hebrew speakers with an operating knowledge of the language, since its inception and development from Paleo-Semitic languages. Rabbinic speakers have been a longstanding tradition within the Jewish religion. It may not have been extant as the official language of any state for a significant period of time, but it was in use amongst that particular group of speakers.

Modern Israeli Hebrew (MIH) has some distinct differences from Biblical Hebrew (BH).

When the Jews came to populate the area with which we are concerned here, they found themselves in a very strange linguistic atmosphere--they had to fill the massive gaps they had in the language of Scripture in order to fit the world in which they found themselves. So they started borrowing words.

BH is an unusual little critter, in that it is a small vocabulary for a complete language, around 6600 words with various forms which arise when parsed in whichever way. When the language was modernized, a very large number of loan-words were imported into the language, many from Arabic, Farsi, Germanic languages, and even English. Thus, we have words like, בַנָאנַה (Banana) and אֹתוֹבּוּס (Autobus/Bus). I cannot recall some of the Arabic and Persian loanwords, but those have been extant in the language for quite some time--proximity to Arabic and Persian speakers has been made evident in the Biblical books of Daniel and Ezra, for instance.

MIH uses the same--or very similar--verbal stem system as BH, with an understanding of the verbal stems which operates in a way as to indicate the syntactic and contextual function of each verb as it appears in the system.

Perhaps more pointedly (or less, as it turns out), MIH has largely done away with "pointed" text as it appears above in the example words I gave. The niqqudot, a system of vowel markings for aided reading pronunciation and clarity or meaning, has been mainly abandoned in MIH documents. All BH is given these diacritical marks in order to ensure the transmission of the received text within its tradition.

If you would like to look further into the field of Biblical Hebrew text linguistics (and Text Linguistics in general) I would like to recommend Emmanuel Tov's wonderful book, Textual Criticism of the Hebrew Bible.

A guy named Eliezer Ben-Yehuda led a revival movement. This movement revised the syntax to be of use in modern times, added loan words to make up for vocabulary discrepancies. The biggest thing was having schools teach the now modernized hebrew to the children that were born there. This meant that children now grew up with hebrew being their main language. Now that a few generations have been born and raised in Israel, hebrew is pervasive. However, most of the street and highway signs are in English, Hebrew, and Arabic.

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