There were several things. One was the Rebbe's innovation of learning Rambam everyday to finish it in 1 years or 3 years.
R. Shach mocked it saying that people already knew of the Rambam before the Rebbe came around, and he said that no one should follow the Rebbe's Rambam learning cycle. (He supported daf yomi, and never rejected it saying that people knew of the gemoro before R. Shapiro). He gave two reasons: that people will pasken from the Rambam and it will disrupt seder. Neither of these points seem to be valid, as people who learn Rambam know that we don't always pasken according to the Rambam; and the Rebbe was very adamant that chitas and Rambam should not be learned during seder.
Another thing of contention was whether you can give land of Eretz Yisroel away. The Rebbe was very against giving away land, while R. Shach supported giving away land for "peace".
The Rebbe got very upset when R. Shach spoke really sharply against non religious Jews and kibbutznikim, saying that they forfeited their right to be called Jews. The Rebbe said that "every Jew is part of God” and that “anyone who berates any Jew is touching the apple of God’s eye. (The Rebbe was always careful to only speak of other Jews favourably, similar to the derech of the berditchiver.)
R. Shach also said that anyone who doesn't believe that the Holocaust is because of our aveiros is a heretic. The Rebbe's opinion on that was
In his writings and discussions on the subject, the Rebbe rejected all theological explanations for the Holocaust. What greater conceit -- the Rebbe would say -- and what greater heartlessness, can there be than to give a "reason" for the death and torture of millions of innocent men, women and children? Can we presume to assume that an explanation small enough to fit inside the finite bounds of human reason can explain a horror of such magnitude? We can only concede that there are things that lie beyond the finite ken of the human mind. Echoing his father-in-law, the Rebbe would say: It is not my task to justify G‑d on this. Only G‑d Himself can answer for what He allowed to happen. And the only answer we will accept, said the Rebbe, is the immediate and complete Redemption that will forever banish evil from the face of the earth and bring to light the intrinsic goodness and perfection of G‑d's creation. (Chabad.org)
The are many other disagreements between them, including about Moshiach, which others I'm sure will add.