The Wikipedia page on De Moivre's Formula says the formula doesn't hold for non-integer $n$, since non-integer powers of a complex number can have multiple values.
It then goes on to say that this does not apply to Euler's formula since:
No such problem occurs with Euler's formula since there is no identification of different values of its exponent. Euler's formula involves a complex power of a positive real number and this always has a defined value
I don't really understand what Wikipedia means. The formula's are very similar, so I would have expected them to have identical properties. Could someone maybe expand on Wikipedia's explanation? Thank you.