The difference is very subtle.
I'm sure that you know 지금 appears in the present tense. 이제 is stuck between the present tense and future tense. With 이제, you speak in a manner implying you're seeing toward the future. When you use 지금, you speak in the present moment.
If you understand the two right, you will have to be able to figure out 이제 갈게, 이제 자야겠다 are softer than 지금 갈게, 지금 자야겠다.