We always capitalize days of the week and months of the year in writing. However, when it comes to seasons of the year, we don’t capitalize. If we did, we would have to capitalize every particular time period, even those occurring throughout the day, like morning and afternoon. This would look absurd and get quite confusing.
Seasons, such as fall, spring, winter, and summer are general nouns, not proper nouns. Days of the week and months of the year are proper nouns instead so they are always capitalized. Therefore, the capitalization that applies to proper nouns isn’t applicable to seasons.
There are situations, however, when seasons are capitalized. For example, at the beginning of a sentence where the rule of the capitalizing the first letter applies (E.g.: Summer is the best season of the year, but fall is not). Other examples of when you can capitalize fall are when the season forms part of a proper noun, or when it’s a part of a title. For example, in ‘Fall Games’ or ‘Fall Semester.’
Another example is when seasons are personified in a piece of creative writing, such as in poems. But this only applies when seasons are given life in such writing.
Overall, the general rule is that we don’t capitalize seasons, fall included.