지금 is used in the sense of "right now" or "at this moment" 이제 is used like "after this time, now..."
For example 지금 눈이 와요- Now, it is snowing (specifically right now) 이제 눈이 와요- Now, it is snowing (after all this time)
I believe that 이제 focuses on changes that happened over time, which led to the current situation. In contrast 지금 focuses on the exact moment something is happened.
This is just what I've learned, I'm not a native Korean speaker. I hoped this made sense.
It rained all day long. And now it started to snow. So you tell Your Korean friend: '이제 눈이 와요' - it snows now (as opposed to the weather before).
Another situation. You go to the shopping mall, with pockets full of cash. After buying lots of stuff, You say: '이제 돈이 없어' - I don't have any money now (but used to have a while ago).
The word '이제' means now, but is specifically used to emphasize some recent change