Brazilian Portuguese vs. European Portugese

I was startled to learn today that there are two ways of spelling the word Portuguese (Portugese).

I am not competent to have any opinion about the significance of this difference in spelling.

Is it just a "Brazil/Brasil vs Europe/European" thing, or is it also related to terms for language vs terms for the people in Portugal?

This issue is debated. See this thread at Duolingo for example.

What are your thoughts on this difference? Should both spellings be viewed as correct, according to context?