자랑스럽다 exactly translates as 'to feel proud' in English. For example, 내 딸이 자랑스럽다. 뿌듯하다 has a sense that you are somewhat overwhelmed by strong feelings of joy and satisfaction about achieving something. For example, 어제 늦게까지 집안청소를 마쳤더니 뿌듯하다.

뿌듯하다 - the feeling that you can feel by yourself after you did nice job 자랑스럽다 - kind of 'be proud of'..?(it doesn't matter whether you did or other people did) I'm not good at writing in English.. but hope you'd understand.

자랑스럽다 only means 'be proud of~' while 뿌듯하다 can additionally mean 'feel great, feel satisfied.....etc depending on context)