지금 and 이제 are both super common words, but speakers of other languages can find them a bit confusing at first if their language doesn’t make a similar distinction and thus translates them both as the same word (English is one of those languages; in English, we would use “now” for both of these Korean words). Let’s take a quick look at the two and how they’re different.
지금 is “now” in the sense of “at this present moment.” Let’s say that you have to catch a flight with your family and someone is holding up the process. Time is ticking away and it’s getting critical. In that case, you would use 지금. You could tell the slowpoke “지금 가야 돼!” (We have to go now!)
이제 is “now” in contrast to some other time. Let’s say that your friend invites you to a concert by an artist you aren’t familiar with and at first you turn down the offer, saying you don’t want to go. Later in the evening, you look up the artist on the internet and find out that their music is your style and you feel that you actually do want to go to the concert. You call your friend and say you want to go, and they ask you to explain yourself. You say,
“가고 싶지 않았는데 가수를 좀 알아봤고 이제 가고 싶어.” (I didn’t want to go, but I looked up the artist and now I want to go.)
In this case, 이제 is “now” in contrast to the time when you didn’t want to go.