Easter is celebrated as the resurrection of Jesus, not his birth. (Source: general knowledge of Christianity)

For info on the date of his birth, I'll defer to experts.

Easter is the celebration of the death of Jesus for the salvation of mankind but more importantly as a celebration of his resurrection from the dead.

His actual date of birth is unknown (I could actually be wrong here) but it was moved to the 25th of December as it was just after the Winter Solstice and so from then on the days would be getting longer/brighter/warmer and so it made sense for the religious powers that be to align the birth of Jesus with the beginning of spring as it represented a light/rebirth/salvation both physical and metaphorical at the end of a cold winter