Hello, Nastassia, and welcome to the Grammar Exchange.
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Both sentences are correct, though you have "always" in an awkward place in "She always is letting me down." You should place "always" after "is" (the auxiliary verb): "She is always letting me down."
There is a difference between the two. The sentence "She always lets me down" merely comments on a behavioral pattern. The sentence "She is always letting me down" speaks of the pattern with more immediacy and expresses irritation.
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