What are some major differences between the G'RA and the Ba'al Shem Tov, in terms of hashkafa, learning style, and other differences? Any sources would be great.
To quote Rabbi Avigdor Miller on this question:
The Vilna Gaon, zichrono livracha, was a follower in the footsteps of his great teachers. He lived by the ideals of the Gemara and the Rishonim in whose footsteps he walked. And he wasn’t interested in making any novelties. And the Ba’al Shem Tov, zichrono livracha, in general also did that. Of course, in scholarship, the Ba’al Shem Tov wasn’t as famous as the Vilna Gaon was. He wasn’t as famous as the Vilna Gaon as a Torah scholar but he devoted himself to an entirely different field! His field was to awaken devotion in the hearts of the multitude. And therefore he used different methods. They weren’t methods that were opposed to the Torah, but they were different methods however.
For instance, the Vilna Gaon had no need to sing a melody. And that’s because he dealt only with the intelligentsia. He didn’t have any direct contact with the masses. He had an influence on the masses, but it was only indirectly. The Vilna Gaon was one of the top scholars, not only of his generation but of any generation. And therefore, he dealt only with great men. Whereas the Ba’al Shem Tov, he dealt also with the general masses. Even with very ignorant people too. And therefore, he had to use those expedients that are necessary to interest even the lower people. And that’s why they lived in two entirely different environments and they had two entirely different forms of endeavor.
Some other points:
Ba'al Shem Tov:
"This is an important rule: Everything in the universe contains holy sparks. Nothing is devoid of these sparks, even wood and stones. There are sparks from the "breaking [of the vessels]" even in all of man's deeds, even in a sin he commits." (Tzava'at ha-Rivash 141)
"…"The whole earth is full of His glory", and that every motion and thought, everything, comes from Him… By means of this knowledge, there is no longer any barrier separating between man and God, and with this all the workers of iniquity are scattered." (see here)
"A generation that has raised its eyes, and spoken words against the most High, This is your God, Israel, in every tree and every stone. And they pervert the verse: Blessed be the glory of the Lord from His place (Yechezkel 3:12), and the verse: And you give life to all of them (Nechemya 9:6). Alas the evil shepherds, who have invented a new judgment and a new teaching, which their students that came after them imbibed, and thus the name of God is desecrated by them." (see here)