Here's why:
* General use: "Spelling bee" is a common noun referring to a type of competition. Common nouns are not capitalized.
* Proper nouns: Specific names of events, organizations, or places are capitalized. For example, "The Scripps National Spelling Bee" is capitalized because it's the official name of the event.
So, you would write:
* "We're going to the spelling bee this weekend."
* "He won the National Spelling Bee last year."