희망

When you use 희망, you mean there is the possibility that the good result or the hopeful future will come. It is the main difference with 소망. 소망 can be used if there will be a good or bad situation. Therefore, it is usually translated as hope in English.

매우 절망적인 상황인 것을 나도 모르는 것은 아니지만 희망을 잃어서는 안 돼.

I am sure that the situation is really hopeless, but we should not lose our hope.

언제까지 지금과 같은 상황 속에서 지내야 하는 건지. 희망이 보이지 않는다.

How long do I have to be in the situation like this? I don't see any hope.

학생들의 희망찬 미래를 위해 수많은 사람들이 장학금을 기부하고 있다.

For the hopeful future of the students, many people contribute scholarship.

소망

소망 can be used for whatever you want. It has the most broad usage. Although it is vague or the possibility that you the dream will come true is too low, you can use that.

그는 아름다운 세상을 만들고 싶다는 개인적인 소망을 이루기 위해 노력하고 있다.

He tries his best to make his personal dream come true which is that he wants to make a better world.

단순히 마음 속에 있는 소망을 이야기했을 뿐인데, 다들 화를 냈다.

I only said the dream in my heart, but everyone was angry at it.

결혼식 날 그 부부는 검은 머리 파 뿌리 될 때까지 사랑하며 살고 싶다는 소망을 이야기했다.

The couple told their dream on their wedding day that they want to live loving each other until the end of their life.