Although using そう with the standard form of the verb rather than the stem looks similar, the meaning is actually very different. Used with the standard form of the verb you are reporting something that you have heard from a pretty reliable source. Consequently, it is often used with ~によると which translates as “according to ~”.

天気予報てんきよほうによると、明日あしたあめそうです。
According to the weather forecast, it will rain tomorrow.
かれはお金持かねもちだそうだ。
He’s apparently very wealthy.

らしい is the same as そう directly above but here your comment is based on hearsay.

かれ新宿しんじゅくらしい
Apparently he’s going to move to Shinjuku.

Context: You heard it from a friend but you can’t be certain.

彼女かのじょはピアノがけるらしいよ。
I heard that she can play the piano.

Context: You haven’t heard her yourself but someone else told you.

If the difference between だそうだ and らしい seems a little vague, that’s because it is. In truth, the former isn’t really used that often and sounds a little hard in conversation. You would like hear it used on the news but less often in every discussions. らしい, on the other hand, is very common.

らしい can also be used to express behaviour, e.g. behave like a gentleman.

政治家せいじからしい態度たいど
Like a politician.
女性じょせいらしくないあるかた
Not a lady-like way of walking.
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