What is the difference between 鸡蛋 and 鸟蛋?
鸡蛋 is a specific term.
鸟蛋 is a general term.

When we know what kind of bird it is, we use a specific term.
Otherwise, we use a general term.

For example, we call the eggs of 鸭 (duck), 鸽子 (dove), or 鹌鹑 (quail) as 鸭蛋, 鸽子蛋, or 鹌鹑蛋.