Dispute between Jews and Christians according to Quran

What was the dispute between Jews and Christians? This is what the Quran says in Surat Al-Baqarah (The Cow) verse 113 (2:113):
The Jews said that the Christians follow nothing (i.e. are not on the right religion); and the Christians said that the Jews follow nothing (i.e. are not on the right religion); though they both recite the Scripture. Like unto their word, said (the pagans) who know not. Allah will judge between them on the Day of Resurrection about that wherein they have been differing (Muhsin Khan translation).
The Jews say: "The Christians have naught (to stand) upon; and the Christians say: "The Jews have naught (To stand) upon." Yet they (Profess to) study the (same) Book. Like unto their word is what those say who know not; but Allah will judge between them in their quarrel on the Day of Judgment (Yusuf Ali translation).

This is the precise truth about the dispute between Jews and Christians:
The Christians said to the Jews, "Jesus is the Messiah (Christ) according to the Scriptures." The Jews said to the Christians, "Jesus is not the Messiah (Christ) according to the Scriptures."

Why can't the Quran be more specific or is Allah, the all-knowing, not aware of the details? Anyway, we do not have to wait till Judgment Day before we know who is right. The Jewish Scripture, which is the Old Testament of the Bible is available on the internet. Anyone who cares can always search the scriptures and determine whether the Christians or the Jews is right. We assure you that the Jews did not corrupt their Scripture just to win. And the same uncorrupted Scripture does not make any reference to Mohammed, despite what the Quran claims (7:157).

Both the Sunnis and Shiites study the same Quran, but they also have disputes, right?