The three words mean SIMILAR things. However there are subtle difference in their meaning and definitions. Couple of minutes reading the dictionary definitions would reveal that. The Jewish, Christian and Islamic faiths are all Abrahamic faiths, so they all share a common basis. The various groups that would one day give birth to what we recognise as the Christian faith were indeed once operated within the Jewish faith. It can't be a sect any longer, as it is separate from the recognised Jewish structures, which is what generally defines the difference between a sect and a denomination. There is no clear separation that would define when a denomination has ceased being such and become its own faith. One might argue that since Christianity and the Jewish faith disagree on several tenant that form the core of their beliefs, that they are no longer close enough in faith in practise to have one associated as a denomination of the other.
I don't think any practicing Jew or Christian would consider themselves to be a part of the same religion. Jews and Christians worship the same God, and hold some of the same texts as sacred. But Jews are waiting for the promised Messiah, while Christians insist that he came already.
On the other hand, Baptists and Methodists would agree that they are denominations of the same religion. They have the same core beliefs, but disagree on certain details, mainly in terms of worship experience (speaking in tongues, specifically). They typically recognize each other as being part of the same religion--Christian in this example.
A sect is a splinter off of a larger religious group. Christianity was considered by the Jews and Romans of the 1st century AD to be a Jewish Sect, until Gentiles started becoming Christians. If we implement the ability to splinter a religion, the feature should be called "Create a Sect". I think a Sect would eventually be considered a denomination, "sect" often applies to new schisms of the faith.
A cult is a small, controlling group led by a single charismatic individual, that has beliefs and practices that are considered bizarre by others, such as polygamy.
Note that I'm providing American English definitions, and that the European words that directly translate to "sect" are what Americans think of as a "cult". So we may need to consider the word choice in the game carefully.
