Both mean "now" but they have slightly different nuances. 지금 means "now" as in "right now". 이제 means "now" as in "now and also the upcoming future" ex) 지금 인터넷이 돼요. The internet is working right now. I'm not sure it has worked or not. 이제 인터넷이 돼요. The internet is working now. It has not worked until just a second ago. 지금 볼 수 있습니다 I can see it right now. 이제 볼 수 있습니다 I can see it now. (I could not see it until just a second ago.)

I got this from hinative.com: http://hinative.com/en-us/questions/15762 지금 한국어 공부해요 = I'm studying korean at the monent. 이제 한국어 공부해요 = now i am to study korean