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After posting my comment to Karawan Kouki's answer, I realized that what I wrote is ambiguous. I guess I better submit an answer of my own.
An Israeli is a citizen or resident of the geographical place that is called Israel.
An Israelite is someone who is a descentant of Jacob/Israel.
Jacob/Israel had twelve sons. Jews are descendants of Judah, who was one of these twelve sons.
Thus, all Jews are Israelites, but not all Israelites are Jews.
After writing my comment to Karawan Kouki's answer, I realized that as I wrote it, it could be interpreted as saying that Jews are not Israelites; that only descendants of Israel's other eleven sons are Israelites. That interpretation would not be correct. In writing “Descendants of Jacob's other sons are Israelites as well, but not Jews”, I did not intend to say that Jews are not Israelites; but that those who are descended from Jacob's other sons — and not from Judah — are not Jews.
Edit in response to comment: Actually, today's Jews are descendants of Judah, Benjamin, and Levi (who did not found his own "tribe," but some of whose descendants formed the hereditary Jewish priesthood line). These are the descendants of the inhabitants of Judea. The descendants of the other tribes, those of the original Kingdom of Israel, have been "lost," hence later references to the "lost tribes of Israel." Here and there, groups will claim a lineage from one of these "lost tribes." The Mormons are the most noticeable example that springs to my mind. However, today's Jews by and large do not recognize these claims. In Jewish tradition, the Lost Tribes of Israel remain "lost". In any case, my Israelite = Jew correlation, as I specifically stated, applies to the time of the Old Testament. The 1:1 correlation is no longer true for the Jews of today. You seem to have missed the very point I was making: that "Israelite" and "Israeli" not only describe different groups of people, but are also to be associated with different time periods. I apologize if the confusion stemmed from any lack of clarity on my part.
Original answer: I would use "Israeli" to refer to an inhabitant of the modern-day nation of Israel (i.e. post-1948). "Israelites" would be the people who lived in the northern part of that territory during Biblical times (the land was divided into 2 kingdoms, Israel in the north and Judea in the south). "Israelite" could also be a catch-all term for all the Jews during Biblical times. Jacob was given the name "Israel" by God, his sons were patriarchs of the "Twelve Tribes of Israel," and these tribes later went on to settle the land which bears his name.
Merriam-Webster's online dictionary backs this up, defining an Israeli as an inhabitant of the Republic of Israel (the modern nation) and an Israelite as either a descendant of Jacob or an inhabitant of the ancient (northern) kingdom (the Biblical kingdom).
Note that not all Israelis are Jews, although most of them are. While the people currently referred to as Palestinians do not have Israeli citizenship, there are some of them who took it earlier on and are now known as Israeli Arabs. They have citizenship and representation in the government. For the Biblical term, though, "Israelite" and "Jew" were pretty synonymous.
An Israelite is a descendant of Jacob (also called Israel).
Jacob:
http://en.wikipedia.org/wiki/Jacob
An Israeli is a citizen of the state of Israel.
Israelis:
http://en.wikipedia.org/wiki/Israelis
http://en.wikipedia.org/wiki/Israel
There is no difference. The people the Bible translators call "Israelites" are the same as Israeli's. Jacob was a man. The Word of God tells us Jacob wrestled with God, and God changed Jacob's name to Israel. Israel's decendents were, are still are Israeli's / Israelites. The reason people don't like to think of today's Israeli's as a continuation of the Biblical Israel is because they see a secular people who fail God all the time. Yet one look at the Bible will show you that the Israeli's in the Bible failed all the time. Indeed, has the Church done any better in 2000 years... no! Thank God for His grace and mercy. Just look at Psalm 105: 10, it states, regarding the Land of Israel, "He [God] confirmed it to Jacob as a decree, to Israel as an everlasting covenant". So what part of everlasting don't you undersatnd? This is a people both then and now chosen, not because they were great, or super holy, or the best, but simply because God chose them... and if you want to argue about that, then you'd better take it to the Lord God and make your case! www.cfi.org.uk

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I edited mine to respond to your comment before seeing your answer! Sorry! But today's Jews include Benjaminites as well!
by Too Much Time On My Hands on April 24th, 2005