계란 is from Chinese roots. 계 means a chicken and 란 means an egg.
On the other hand, 달걀 is a pure Korean word.
닭 is a chicken and 알 is egg. The ㄱ moved over in 닭알 to form 달걀.
Also 달걀 also does not mean cooked eggs, 계란 works for cooked or uncooked.
달걀 and 계란 are exactly the same. You can refer to either with both. Though the National Institute of the Korean Language recommends us to use 달걀 instead of 계란, you are unlikely to hear any Korean native speaker say 달걀.
Then would you recommend that we forget about 달걀 and just know it simply exists rather than to study it and keep it a part of our korean vocabulary?
Wasn't egg jidan in chinese??
jidan()@#Q$((#Q$&^(#4#Q(%&(*and boom gyelan is the new word
Although we actually say "ji1 dan4" 鸡蛋 in Mandarin, the closer analogue here is "ji1 luan3" 鸡卵.