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
While it's partially because of the robotic Duolingo voice, Korean consonants (especially nasal sounds like 'N,' 'M,' etc.) are pronounced a bit differently than English. I can't explain it well myself but I recommend watching this video if you want to know about the pronunciation differences.
Korean has a tendency to express continuous action of no special importance, like Japanese, with present tense. (Not to mention the occasional future action.) This might just be because it's terse and easier to say quickly, or it could be that the ambiguity is preferred in some cases. Present continuous is really used for extra emphasis in Korean. Like, for example:
A:있잖아 내가 준 그 프로젝트는? So, how about that project I gave you? B: 하고 있어요. I'm doing it.
In that case a "해요" might not be enough, as it's more ambiguous, and it's unclear whether you're now starting or have already started, and therefore kind of inappropriate. I think the present tense substitute for present continuous happens when it's clearer, with one's own eyes, what's happening and therefore requires no explicit differentiation.
A: 눈이 와
B: 응? 지금?
A: 응. 봐봐