Sometimes giving Simplify the optional second argument, which are the assumptions that you are telling Simplify that it can depend on always being true and which it is free to make use of as it sees fit, can accomplish interesting and helpful things.
Simplify[a^4 - 4 a^3 z + 6 a^2 z^2 - 4 a z^3 + z^4, a - z == y]
immediately returns
y^4
