@Broncon007,

To my knowledge, the usage of 发觉 is much closer to the English verb, "realize" and 发现 is more like the verb, "find" or "discover".

EG: 哥伦布发现了新大陆。

Columbus discovered a new continent.

发觉 cannot be used in that way, you couldn't say, 哥伦布发觉了新大陆 (Columbus realized a new continent).