all is one option:
> A <- c("A", "B", "C", "D")
> B <- A
> C <- c("A", "C", "C", "E")
> all(A==B)
[1] TRUE
> all(A==C)
[1] FALSE
But you may have to watch out for recycling:
> D <- c("A","B","A","B")
> E <- c("A","B")
> all(D==E)
[1] TRUE
> all(length(D)==length(E)) && all(D==E)
[1] FALSE
The documentation for length says it currently only outputs an integer of length 1, but that it may change in the future, so that's why I wrapped the length test in all.