Apollo suggests that the trial will stand as a permanent resolution to the problem and thus rid Orestes of his pursuers, rather than simply being a temporary sanctuary.
The Furies are opposed to the trial for several reasons. First, they distrust the younger gods and see a trial run by them as rigged. They do not feel that the younger gods should have the power to try them, and regard the trial as a form of disrespect to their ancient powers and roles. Next, they say that a trial which might end up letting a matricide go free is a mockery of all that is right or moral.
The Furies distrust Apollo and Zeus because they feel that they are usurpers of the rightful power of the older chthonic deities. Also, Zeus himself was a patricide and Apollo, following his commands is supporting matricide, and thus the younger gods are overturning the order in which the younger generation should honor, respect, and obey an older one.