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.