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If I'm not wrong, そうだ is used when you hear something from someone else and you are trying to relay it forward to another person. For example, you hear なつみさん say that 高田さん is sick. So when the teacher asks you wheres 高田さん、 you reply: "高田さんは病気で休んでいるそうです". But what about ようだ? What if you replaced it and said "高田さんは病気で休んでいるようです"? What difference does it make?

Ive also heard that ようだ is used when you notice something. For example, you see your friend 若宮 laughing with your elder brother, and you are shocked. You silently think to yourself: "若宮さんは兄貴と仲がいいようですね..." Similarly, does it make a difference if i replace ようです here with そうです?

All in all, what are the differences and similarities between ようだ and そうだ?

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I often had this problem myself when I was translating some manga and sometimes I still do. Based on my encounters with these words, I also know that when you use
そうだ it means that you are talking/relaying about something based on what you heard from other sources, just like in your example above
(You have heard that apparently Takada-san is sick). I would often had problems distinguishing this そう with the one that means "it appears", "looks like").

As for ようだ, I think it's used when you are witnessing/observing something at the moment, just like you said in your example, you observe your brother and friend laughing and they look as if / seem like to get along based on what you are seeing and deduce from the event.
Perhaps you also confused sometimes the そう of hearsay and the one that means "looks like" such as in the example: "This dress looks expensive" このドレスは高そうだ (or です depending how polite you want to make the sentence).
I hope I didn't make things more confusing, but I encountered the same things as well and I wanted to share my experience with these structures as well.

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