The Nicene Creed in use today does not date from the First Council of Nicaea in 325 but from the First Council of Constantinople in 381. They are similar but there are a number of differences. The Western churches made a later insertion into the 381 creed, adding Filioque ("and from the Son") to the original text.
It is a product of both councils (as well as some later refining). That is why it is more accurately called "the Nicene-Constantinopolitan Creed." "Nicene Creed" is just kind of shorthand since no one wants to go around saying "Nicene-Constantinopolitan Creed."