They are often used to mean the same thing, especially in the C of E.
Strictly though, the christening service is for a child, when their parents and godparents undertake to bring them up as Christians, it includes baptism into the church. Adults make their own promises when they are baptised. That is: all those who belong to Baptist churches, and anyone in the C of E who presents themselves as an adult to be baptised in to the church.
I don't know about other churches' terminology.