The later language of the Mishna, Talmud, and Midrasha comprise Talmudic Hebrew.2 Their forms were more adaptable to everyday life, as they themselves dealt more with everyday life. They were considered by some to be perversions or corruptions of Biblical Hebrew, though, and so there was some resistance to their use.
In medieval Europe and then later during the enlightenment, Hebrew was used in a written form as a scholarly and literary language, much like Latin. Writers wanted to use pure Biblical Hebrew, but its limited vocabulary was often restricting. Writers would use words from Talmudic Hebrew, take loan words from the local vernacular, and build new ones off old roots3. These expansions were denounced by some as blasphemous, and some even objected to the application of Hebrew to secular affairs, but they gave the later revivers precedents. The words introduced here were generally useful, but there was much variation among writers so many useful terms never made it into Modern Hebrew.
The Jews of Yemen also had their own form of Hebrew, predictably called Yemenite Hebrew. It is considered to be the most phonologically conservative of the surviving varieties as it has a unique sound for each consonant save one. It was not, however, used as a source for Modern Hebrew. This may be because Ben Yehuda and others did not know about it or perhaps that they did not properly examine it, but regardless, it was not used.