(NB NOT Karaimism)
Karaite Judaism is the 20th Century culmination of centuries of Jewish opposition. It was crystallized by an East European Karaite Turkic Judaizer, Tuvia Levi Babovich, who moved to Cairo to save the ignorant and decrepit communities of Egyptian Qaraites from oblivion by merging his tradition with theirs. Although Tuvia brought with him some Crimean Karaite followers from Eastern Europe, under the influence of Mordechai Alfandari the new religion came to be known as Karaite Judaism even after the last of the original Karaites in Tuvia's movement had passed away. It survives today thanks to the Universal Karaite Judaism organization based in Israel.
It is sometimes proposed that that the modern state of Israel (which was founded on the back of the Subbotnik migrations) could operate better as a Karaite Jewish state and this has led to a massive concerted effort to deny the Subbotnik origins of Tuvia Levi Babovich in order to increase Karaite Judaism's appeal as an alternative for reform Jews whose religion is not recognized by the state of Israel.
For more information about Karaite Judaism, please see Meir Rekhavi's Karaite Jewish University which runs under the auspices of Universal Karaite Judaism.
The New "Karaite" Jews
Karaite Judaism was discovered by Mordechai Alfandari, an American Talmudic Jewish apostate of Turkish origins who was attracted to the proposition of rejecting all Oral Torah like a Sadducee and saw Tuvia's Karaite Judaism as the possible platform upon which he could achive his dream of resurrecting the Sadducee religion. Mordechai was only posthumously accepted by the Karaite Jewish community for his efforts and allowed to be buried in the Egyptian Qaraite Jewish graveyard in Israel. His "World Karaite Movement" was continued by Meir Rekhavi who opened the Karaite Jewish University in association with HaYahadut HaQarait HaOlamit of Moshe Firrouz cleverly mistranslated as "Universal Karaite Judaism" rather than "The Universal Qaraite Judaism" thereby blurring the once significant distinction between Karaite Judaism and Qaraite Judaism. Many more Karaites as well as Karaims (such as the composer Abraham Kefeli) have been absorbed into Yehudim Qara'im of Israel as Karaite Jews since then.
The Neo Karaite Jews are people who adopted Mordechai Alfandari's Karaite Judaism but objected to Meir Rekhavi uniting it with Moshe Firrouz's Universal Qaraite Judaism.