Well, you have to ask the distributor/producers of the US version of the film. I was merely quoting the Wikipedia page.
They are assuming that the cost of the rights to that song was just too expensive for the budget the distributors had had for the US release of the film.
Another possibility is that Ms. Brightman's "people" had reservations about her song being featured in a kids' anime movie released in US (a much larger market compared to Japan), where she may have an public image that they would like to maintain and thought this may somehow do damage.
For example, many big-name hollywood stars appear in TV commercials in Japan. And most of them prefer not to have that known in The US. Why?
Let us take the case of Jodie Foster as an example. She is an accomplished, much-respected actress/director/producer in her native country. Appearing in TV ads for cars or canned coffee could possibly "cheapen" her image as a serious actress. She does it for Japanese TV because the ads are only shown on Japanese TV (and it pays well).
I am probably reading too much into it but when it comes to business like this, there are so many factors to consider and they just have to do what they have to do. In this case, they had to, or decided to replace Ms. Brightman's song with another song.