As Aaron Christianson has pointed out, script and language are not the same thing: the Palaeo-Hebrew script is one thing, and the language was another—just as the Roman script is distinct from, say, the English or French or German language.
The Palaeo Phoenician/Hebrew script was used by all Canaanite nations in from about 1500 to 500 BCE. The Hebrew language of that period is commonly known today as biblical Hebrew—and that bears much the same relationship to modern Hebrew as Shakespearean English does to modern English.
The Palaeo Hebrew script differs from the later Square Hebrew script (which was based on the Aramaic, which itself was derived from the Canaanite), only in the shape of the characters. The practice of writing only in consonants was the common practice in both scripts, and people were able to read it much as you and I are able to read abbrvtd SMS txt wrttn wth cnnsnts nly.
It is only in modern times that ktiv malé (“full spelling”) mode has been introduced in Hebrew, to indicate the /i/, /o/ and /u/ vowels.
EDIT: Aaron has correctly pointed out to me that full spelling was in fact frequently used in the Dead Sea Scrolls and in the Mishnah period (ca. 50–220 CE).
(Originally written on May 21, 2016, in reply to a question at Quora.com).