Which sentence is correct (if not both) and what is the difference between them?
- I just took a bath. [moments ago]
- I've just taken a bath.
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Both sentences are correct. "I've just taken a bath" is only adding the word "have" into the sentence.
In my idiolect (and that of many older British speakers) I would not use the simple past with "just", so I would only ever say "I have just taken"* But the form "I just took" was common in American English when I was young, and has now become more common in British as well.
But among those who would use either, I don't think there is any difference in meaning.
*On a separate matter, the British idiom is "have a bath" rather than take a bath", so I would say "I have just had a bath" rather than either of your two sentences.