MtDNA is passed from the mother to all children. That's why everybody can test for it (unlike Y-chromosome DNA which only men have). It can be used to trace the maternal like nevertheless because mtDNA is not on the 23 chromosomes and does not recombine.
Y-DNA and mtDNA haplogroups are always different. They are completely separate things. You can see the evolution of both Y-DNA and mtDNA haplogroups on this chart.