Zygote is the first stage in the development process of an organism. It is produced of the process of fertilization occurring amongst the haploid ovum cell from the female and the haploid sperm cell from the male.
Embryo refers to that stage in the developing process of the organism which starts from the initiation of the first cell division till the birth
The mitosis of the zygote results in to the formation of the embryo. So zygote is the first stage in the development of a new organism while embryo is the stage that follows next.