What is the difference between "أُحْيِي" (I give life) and "تُخْرِجُ" (You bring forth/cause to emerge) in the context of raising the dead, and why was the word "تُحْيِي" (You give life) not used instead of "تُخْرِجُ"?

The difference between the expressions "(I) bring the dead to life" in Surah Al Imran and "you bring forth the dead" in Surah Al-Ma'idah lies in the context of each verse:

In Al Imran, the verse came in the context of glad tidings about Isa (Jesus) and introducing the Children of Israel to his prophethood. Isa ascribed the power to Allah, saying, "and I bring the dead to life by Allah’s leave."

As for Al-Ma'idah, the verse came in the context of reprimanding the Christians, so the attribution in it was from Allah to Isa, as a grace from Allah to him. The word "bring forth" was used to indicate Isa's ability to resurrect the dead after their burial and bring them out of their graves. The repetition of "by My leave" four times in it is to emphasize that all these miracles are by Allah's leave and power, and not by Isa's independent capability.