Gözün aydın is used when something long expected happens. Like when a baby is born (to the parents or grandparents), someone returns from military service (to the family), someone returns home from a long journey etc. "Tebrik ederim" is very similar to the usage of "congatulations" (direct translation being i congratulate), like two people get married (to the couple), someone gets a degree or achieves something (to the achiever).
Take care. :)