They were contemporaries and similar in many ways.
Hi Yisrael and welcome to Mi Yodeya.
Both Rabbis embraced the notion of 'Torah Im Derech Eretz' and cultivated what has since been coined 'Neo-Orthodoxy'. They also both embarked on a policy of Austritt or 'Withdrawal' leading separatist communities (Rabbi Hirsch's Israelitische Religiongesellschaft (IRG) congregation in Frankfurt and Rabbi Hildersheimer - the Adass Jisroel community in Berlin) in a bid to fight against the emerging Reform movement and non-observant element in the community.
However, it is this area of separatism where they differed. There is a very astute critical analysis here where the author Yisrael Kashkin notes that Rabbi Hirsch's secession was far more comprehensive whereas Rabbi Hildersheimer was willing to work to a degree with the non-observant members of the community.
Indeed this view is supported in a biographical account of Rabbi Hildersheimer here where it states:
What can be said is that Hirsch was the more conservative of the two: He was more of a segregationist, who had no interest in cooperation with his ideological opponents; more of an instrumentalist with regard to secular studies; and by no means sympathetic to the emerging nationalist strain of Zionism.