Did the Karaites and the Sadducees believe in the same "Judaism"?
The Ramabam in hilchos Teffilin 4:3 says that if one wears teffilin on his palm or head teffilin on his forehead then he is doing the practice of the Sadducees. The karaites never wore teffilin at all so that seems to be one difference.
The Karaites claim no connection to them since the Sadducees lived in Judea and they lived in Egypt also the Sadducees lived 700 years apart from the karaites since they ceased to exist. The Karaites claim to hold the same philosophical approach of the Written Torah. They also claim that don't know alot about them to begin with.
Just to add there were different groups(betusim,essens,annaim) that only believed in the written Torah but they all seem to have a different twist on interpretation(some more literal some more liberal in translation)
The Sadducees and Karaites are distinct historical movements, though similarities in their approaches to Tanakh suggest they may be related in some sense. This Karaite website, answers this question with
We know very little about the beliefs of the Sadducees. We do know however that like the Karaites they believed only in the authority of the Tanach and rejected the Oral Law.
Another important difference is that the Sadducees have ceased being a movement centuries ago, whereas Karaites continue to be a functional sect today.
Well, it depends who you ask.
Rabbi Avraham de Cologna wrote a kuntress about Karaites, called "Kara Ha'gever". In it, he explains about the history and beliefs of the Karaites. According to him, the Karaites are the spiritual descendants of the Sadducees:
"...ומאז יצאו אלה כתות הארורות של מינים, ויקראו בארץ מצרים ובקוסטאנדינא ובקרימאה ובליטא במלכות משך בשם קראים: ושמותם אצל החכמים צדוקים ובייתוסים, והם אשר התחילו להשיב על הקבלה ולפרש הפסוקים כפי מה שיראה להם מבלתי ישמעו לחכם, הפך אמרו יתברך (על פי התורה) על פי התורה אשר יורוך ועל המשפט אשר יאמרו לך תעשה לא תסור מן הדבר אשר יגידו לך ימין ושמאל: כל זה הגיד לנו האיש משה גדול מאד הרמב"ם [ז"ל] בפירושו למשנה ג' במ"ק לאבות, וזה יספיק לנו ללמדנו כי התחילה כת הקראים לפרסם [שמה] בימי חכמי המשנה וראשיהם היו צדוק ובייתוס תלמידי אנטיגנוס"
Translation: "...and since then came these accursed sects of heretics, and they were called in the land of Egypt and in Constantinople and in Crimea and in Lithuania in the Kingdom of Meshech by the name of Karaites. And their names by the sages Sadducees and Baytusees, and they are the ones who began to retort on the tradition [kabbalah, i.e. the Oral Torah] and to explain the verses as they saw fit without listening to the sage, the opposite of what The Blessed [Hashem] said (according to the Torah) You shall act in accordance with the instructions given you and the ruling handed down to you; you must not deviate from the verdict that they announce to you either to the right or to the left. All of this was told to us by the esteemed man Moshe the Rambam z"l in his commentary on the Mishna third [mishna] of the small tractate Avot, and this is enough to teach us that the Karaite sect began to make itself known in the time of the sages of the mishna and their leaders were Tzaddok and Baytus students of Antignos." (pg. 6)
Rabbi Bernard Revel similarly wrote in "The Karaite Halaka" that this view was that which was held by several Geonim and Rishonim:
"On the other hand, most of the Mediaeval Jewish scholars seem to agree that Karaism was due to a revival of the Sadducees (Abraham Ibn Daud) or that Sadducean elements are prominent in it (Saadia, Juda Halevi). Saadia Gaon...add that the remnants of Zadok and Boethus joined Anan...about two centuries later...the three great lights of Toledo, Judah Halevi, Abraham Ibn Ezra, and Abraham Ibn Daud...and they all assert that Karaism is an offshoot of Sadduceeism...likewise, Maimonides...adds: "In Egypt they are called Karaites, while in the Talmud they are named Sadducees and Boethusians." (pg. 6-8)
Rabbi Revel himself, however, disagrees with this view. He believed that the Geonim and Rishonim held this view out of a need of stamping out what was perceived to be a theologically-dangerous sect of Jews. In reality, as he explains throughout the book, there are vast differences between the Karaite and Sadducee Torah views and in fact, according to him, the Karaite views are much closer to the little we know of the halacha of Philo and the Jews of Egypt, and in many ways, they even agree with the Pharisees. As he writes:
"The only agreement between the Sadducees and the Karaites known to us is their rejection of "water libation", נסוך המים, on the Feast of the Tabernacles...Schechter has established close relation of "Fragments of a Zadokite work" discovered and published by him...with the כתאב אלצדוקיה [Kitab AlZadokiyah] mentioned by the above-named tenth century Karaites. As Shechter himself says: "The term Zadokites naturally suggests the Sadducees; but the present state of knowledge of the latter's doctrines and practices does not offer enough points of resemblance to justify the identification of them with our sect"..." (pg. 44-45)
The book is dedicated to analyzing the differences and similarities between the Karaites and the others sects based on their respective sources.