Mithra celebration was on Dec. 25th. The Christians found it convenient to celebrate the birth of their own god at the same time. From what I have read, that is where the similarity ends.
It is very hard to demonstrate a causal relationship between Christianity and any other specific religion. All the religions of the time had similar components and were offshoots of each other. The Christianity we see today was nothing at all like Christianity of ancient Rome. Bart Eherman asserts there was a greater variety of Christian faiths during the first century that there is today. It took over 300 years for all the faiths to fade away and for a few of them to become dominant. Correlation is not causation.