I always thought that "Jew" was "eврей", but I have also heard the usage of "иудей". What is the difference? They appear to be the same in the dictionaries that I have checked. Are they completely the same, or do they have slightly different meanings/connotations?
Are "eврей" & "иудей" completely the same, or do they have slightly different meanings/connotations?
They are not the same. They are from different categories.
"Иудей" stands for "Judaist" - http://merriam-webster.com/dictionary/Judaist That is, religion.
If you look for "almost the same with different connotations" words, denoting ethnicity, then those would be the pair of "Jew"="жид" and "Hebrew"="еврей".
In modern Russian "жид" is considered a vulgar slur, while in 19th century Russian and in modern Polish "жид" is a neutral "technical" term.
- In political category there is yet another word "Israeli"="израильтянин" - a citizen of the Jewish state.
Now, in essential language those three categories sometimes get blurred - either by mistake or intentionally - and those words end being used as substitutions for one another. The vocabularies you consulted could represent those blurry lines by including those tangent meaning into the articles. However in the strict meaning those words are represent historically related but different attributes.
PS. Etymologically probably "жид" is a simplified rough translation of "Jude", which itself probably stems from "Judaist"/"Иудей". Back then few centuries ago the concept of ethnicity was weak if existing, and people were mostly split over religious watersheds, thus there was little if any separation of "religious" and "national" meaning of those words back then. Again, the word "жид" was mostly used in prior centuries, so could denote both senses at once. So I guess in reading texts using that word it would not be easy to choose between competing interpretations. However in modern Russian this word is only used in some specific circumstances and the "normal" words "еврей" & "иудей" have their clear distinction of concerns.
It's quite simple: "еврей" is nationality or ethnicity (not sure which fits better, but hope you get the point) and "иудей" is someone who follows Judaism.