First, Jews do associate themselves with both terms. "Jews" is a term derived from "Judea" (which itself is via Latin--you may also recognize the slightly more direct Judah, or Yehudah in Hebrew), and obviously today that's the term Jews use for their religion. Not in every language, but English and Yiddish for starters, and those are two big ones. So perhaps the question should be why use either term in general.

Anyway, the term "Israel" is actually a somewhat more common one. In the bible the people are usually called Israel, with Judah being a specific tribal sub-group. Over time, there was a political split between the kingdom of Judah and of Israel, and Israel was conquered by the Assyrians and eventually the term Yehudi came to refer to all Jews. The first reference (that I know of, anyway) is actually a relatively late biblical book, the book of Esther. Part of it is that as the last political entity was Judah, the following ones were too, from the Persian Yehud Medinata to the Roman Iudaea.

But the term Israel retained currency. In Rabbinic texts the term Israel is usually used, from Am Yisrael (the nation of Israel) to shor Yisrael (an ox belonging to an Israelite). In modern English we separate the terms Israelite (a member of the ancient religious group, or less often one of its descendants) from Israeli (a citizen of the modern state) from Israelian (relating to the ancient kingdom of Israel, as opposed to Judah). The term for someone who is does not have priestly or Levite lineage is called Israel. In the formula annoucing a child's name the formula is vayikra shemo b'yisrael, "his name will be called in Israel...". Etc, etc, etc.

Using Jew in English is mostly because European languages (including Yiddish) got it (indirectly) from Latin, and they referenced the name of the province of Iudaea. But there's no real ideological reason Jews couldn't call themselves "Israelites", it's just not convention. Incidentally, there have been other terms used. It was once fashionable to call Jews "Hebrews", since "Jew" had taken on negative connotations. This can sometimes be seen in the names of older organizations, like the Hebrew Free Loan Society, or George Washington's letter to the Hebrew Congregation of Newport. This is actually the norm in Russia--the normal term for "Jew" is "Ebrey" (from "Hebrew), while the Jew-derived "Zhid" is a slur. Of course in modern English some people find "Jew" to be offensive, but they only use "Jewish", rather than another term entirely. Apparently in some places the Yiddish "Yid" is a slur, even though in Yiddish it's neutral or positive.

Anyway, the tl;dr is that Jews usually use terms based on Judea, like Jews, Jewish, etc, which is what you called them in your title, which I'd say is pretty damning evidence! But in Jewish sources and texts Israel is used plenty too. They're both terms that are used, and it just so happens that today Judah-derived terms are used in English, and Israel is used for the political entity.