There is currently no evidence to show that MRI used for routine breast cancer screening saves lives. MRI works differently than mammography and some might consider it "better" because it is unaffected by breast density and does not use radiation.
However, when used for screening, MRI yields even more false positive results than mammography. This is one reason why the use of MRI screening is not recommended for the general public. Even among those at high-risk for developing breast cancer, one study of women with genetic predisposition found that the number of unneeded biopsies tripled. Current studies are looking at the impact of using MRI alone or in combination with mammography for screening high risk women, but it will be many years before we know if the technology saves lives.
As with any screening tool, the potential risks and benefits need to be carefully considered.