When Chassidus came about, it was very controversial. The Vilna Gaon even put Chassidim in cherem. What were the reasons for the controversy? And why is there peace and mutual respect today between the Chassidim and Misnagdim?
Hasidism was not 'controversial.' There indeed was a long battle between Hasidism and the Misnagdim (at the level of street battles and burning of books) but the polemic was just as much about miscommunication as an actual Halakhic difference. So 'controversial' is the wrong word.
The Vila Gaon (GR"A) was personally opposed to Hasidism, because he believed them to be heretics. Why did he see them in this way? Because they presented a new and innovative form of religious practice. In particular the new forms of prayer (such as somersaulting) and focus on worldly pleasures. Indeed, the Gaon himself was a Hasid-- he was referred to by his disciples as "Ha-Gaon He-Hasid." But he was a different kind of Hasid, related not to Hasidism as we know it today but the medieval Haside Ashkenaz, who were Mystics but focused on (1) asceticism and (2) elitism. The challenge that Hasidism presented was not a new Kabbalistic belief, but its popularization.
The relationship with Sabbatianism: certainly the belief that Hasidism was related to Sabbatianism was due to the fact that the Hasidim were threatening, and as the Jews who would become known as the Misnagdim didn't know much about Sabbatianism except that it was related to mysticism would have confused them and think that there is a causal link. However, just as the Vilna Gaon was a mixture between the concept 'Gaon' and 'Hasid,' so we should also think about the Besht. His beliefs were not the radical Hasidic ideas. The concept of the immanent God (that God is in everything) only reached its apex in the thought of Shneur Zalman of Łiady. The Besht, indeed, was a combination of the new Lurianic ideas but also the old concer of the mystical Baal Shem (a folk healer -- tied to popular religion and superstition). That is also a Kabbalistic idea, but tied to the "practical" Kabbalah of Abraham Abulafia (13th century).
The battle between the Hasidim and Misnagdim was only resolved after the death of the Vilna Gaon. In the early 19th century two things happened: first, the Russian government acknowledged the hAsidim, and the Hasidim's attempts to "conquer the communities" were highly successful. It really was not related to opposition to the Haskalah -- though the Misnagdim and Hasidim did come into an alliance against the Haskalah in the second half of the 19th century. The establishment of the great yeshiva of Hayyim of Volozhin (1802) created a new kind of environment where Hasidism and Misnagdism could meet. But apeven so there still remained a great religious divide between Misnagdism and Hasidism. Misnagdism essentially believed in a dualistic religious existence -- that this life is transitory to the next, and that we suffer here for the future world to come. Ultimately it was a very pessimistic view on the possibility of human salvation. The Hasidim, on the other hand, believed in a monistic world -- I.e. that one should enjoy the current life, and that salvation (devekuth or mystical union with god-- in medieval terms, unio mystica) is available to the average person.
Depending on who you ask, you may get slightly different answers. This is what I understand, from a mostly historical and Chassidic perspective. (Garnered from R' Yossi Paltiel1, Berel Wein2, and a bit of Wikipedia3.)
The Baal Shem Tov started his teaching not long after the false messiah, Shabtai Tzvi created massive chaos in the Jewish world. Shabtai Tzvi, among many other things, was a kabbalist. The Baal Shem Tov (and the Baal Shems before him) were therefore regarded as highly suspicious due to their teaching kabbalah. Many of their kabbalistic practices and beliefs, although more common among Sephardim (since kabbala was always more accepted among Sephardim), were very foreign to Ashkenazim, and considered heretical. Also, even Misnagdic Kabbalists (e.g., the Vilna Gaon), disagreed about different kabbalistic teachings that the Besh"t taught. (Even though, AFAIK, the Baal Shem Tov's teachings all had strong basis in Lurianic Kabbala.)
Edit: It is very important to note that the above reasons aren't really the things that Misnagdim disagree on with Chassidim. The Misnagdim did, and do, have ideological disagreements. The above reasons perhaps can explain why the Misnagdim were so suspicious and rejecting of Chassidus, when they otherwise might have just disagreed with, yet accepted, it. See comments below.
The reason for the peace that (B"H) exists now is due to several different reasons.
First, much of the dispute between Chassidim and Misnagdim (at least after the Maggid) existed in Lithuania and White Russia. Ukraine and Poland were already very Chassidic.
The leading dynasty of that region (to my knowledge) was Chabad. The Alter Rebbe did his best to make sure that there was no negative response from his Chassidim to Misnagdic opposition. His son, the Mitteler Rebbe, made the divide even greater. He told his Chassidim to avoid dealing with Misnagdim at all. Don't daven in their shuls, etc. This allowed tempers to cool, and by the end of the Mitteler Rebbe's leadership, much of the old animosity had disappeared.
Then came the ultimate uniting factor: a common enemy. In this case The Enlightenment movement (not to be confused with modern reform, btw). The Enlightenment did their best, through the governments and other means4, to forcibly intellectualize Judaism, at the expense of Halacha. The Misnagdim recognized that Chassidism was not what they had thought it was, and the Tzemach Tzedek and R' Itzhele Volozhiner had a close relationship. Together they worked to prevent the government from forcing secular education on orthodox Jewry. (Crucially, this was not that they did not want Jews to be educated in secular subjects, but that it should not come at the expense of Jewish practice and Jewish education (Jewish Life and Learning).)
By this point, the Geonim among the Misnagdim had realized that Chassidus was not heretical, and did not pose a danger as Shabtai Tzvi and Yaakov Frank had, and so the reason for the original disagreement fell away.
If you disagree, or have a correction, please do correct me!
1 Who presents (mostly) the Chassidic perspective.