Basically they have the same meaning, 'more', but they are different in the usage. 더 ; 밥 더 먹어라. 일을 좀 더 해야해요. 돈을 더 벌어야 겠어요. 더욱 ; 더욱 착한 사람이 되어라. 더욱 바빠졌다. So, you may use 더 as adjective, 더욱 as adverb. or I will show you my Korean English Dictionary 더 힘드는 일 (much) harder work. 더욱더 more and (yet) more. 더 많이 much[a lot] more(양); (a good) many more(수). 하나 더 많다 be too many by one; be one too many. 돈을 3만 원 더 지불하다 [얹어서] pay 30,000 won extra; [잘못하여] pay 10,000 won too much. 한 발짝 더 나아가다 go a step further; take another step. 더 생각한 뒤에 after further consideration. 더 주십시오 Give me some more. 더 할 말은 없다 I have nothing more to say. 그는 그 일에 대해서 더 생각해 보고 싶다고 말했다 He wanted more time to think it over. 더 좋은 것으로 주시오 I want something better. 너도 키가 꽤 크지만 그는 더 크다 You are tall enough, but he is still taller. 더 있어도 되잖아 You can stay a little longer yet, surely? 더욱 더욱 힘드는 일 (much) harder work. 그래서 더욱 the more…because[as, that]; so much the more because; the rather that[because]. 더욱 좋다[나쁘다] be so much the better[worse]. 더욱 중요한 것은 what is more important. 더욱 노력하다 make greater[further] efforts. 더욱 많은 사람을 고용하다 engage still more[much more] persons. 더욱 좋게 만들다 make ((it)) even better. 더욱 열심히 공부하다 work still harder. 난처한 데다가 더욱 난처하게 만들다 [원인이 주어] put the coping stone on one's embarrassment. 그것은 더욱 곤란하다 It is still more difficult. 그는 학문도 없지만 경험은 더욱 없다 He has no scholarship, to say nothing of[not to speak of, much less] experience.

I believe 더 means 再 in Chinese, while 더욱 means 更.

I believe 더 means 再 in Chinese, while 더욱 means 更.