Same reason we call any group anything, really, convention. There's no inherent reason we couldn't use the same term.

The difference between them is the etymology. Israelite comes from the Greek Ἰσραηλίτης Israēlitēs, which is used as a term for...Israelites in Greek biblical texts. They also often use the simpler Ἰσραήλ, but both terms can be found in biblical Greek lexicons. English got this term via Greek.

The term "Israeli" is much more modern. It's essentially a transliteration of the Hebrew ישראלי, which adds an -i suffix to "Israel". It's a common derivation of adjectives and demonyms in Hebrew, and is often used in English (cf Iraqi, Afghani, Pakistani, Kuwaiti). Interestingly Yemen has a similar doubling, with Yemeni and Yemenite.

If that wasn't confusing enough, there's a third term that's rarer--Israelian. It's used for things pertaining to the ancient northern kingdom of Israel (as opposed to the southern kingdom of Judah), as opposed to the Israelites as an ancient ethnic group or modern Israelis. There are papers written about Israelian Hebrew in the bible, for instance.