Flare, which is both noun or a verb (the verb usually followed by up), almost always has to do with fire, though it’s sometimes used metaphorically, as in the phrase tempers flare. The other exception is where it refers to a rounded, spreading shape, as in a piece of furniture or an article of clothing.
Flair refers to either (1) distinctive elegance or style, or (2) a natural talent or aptitude. For example, one might dress with a lot of flair, or one might have a flair for writing poetry.
The flare will come and go in a few seconds and will get brighter than Venus in the evening sky. [Mankato Free Press]
Like Novotny, Streever is a scientist with a flair for anecdotes. [Powell's Books]
Wildfires flare up in western valley [Times-News]
The July 4th weekend is notorious for boaters shooting off their emergency flares as a substitute for fireworks. [KHON 2]
The fair included free food, games and activities with a Western flair. [NewsOK]
Flare-ups caused by the heat and wind on Thursday were quickly doused with the use of helicopter water buckets and water bombers, provincial officials said. [CBC News]
A white cropped sweater bearing the midriff and high-waisted fit and flared skirt looked positively fresh. [NYT]