Hi! Here's a quick way to list the minor difference between them: The original way of mentioning days of the week is by using 曜日 as in 月曜日、火曜日、… A more easy and abbreviated way would be using 曜 as in 月曜、火曜、… A more detailed explanation: First you have to know that what 曜 exactly refers to is "planet" in astrology. Thus, 月曜 means "planet of Moon" and 月曜日 is "day of planet of Moon". So it's like we're actually calling it "Moon's day" when we say 月曜日, because 日 means (day) in some instance.

Hi! You don't have to worry about that. We don't see distincion of both, and just use the same way. However, birthday is 「誕生日」, and you cannot say 「誕生」, which never means the same. Happy Monday! 楽しい月曜日を！