I just want to know what the difference is between the two and their proper usage because in dictionary they seem almost the same.
The difference is minimal. They can be used interchangeably most of the time.
The actual conceptual difference between 月曜 and 月曜日 is not so hard to understand, either. 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, and "the Moon's" for 月曜.
This situation is somewhat similar to the word "English". When we say "English" alone it's usually understood as "English language" (or "English people"). But it doesn't mean "English" automatically stands for the language, as you couldn't rephrase "English muffin" as "English language muffin". The same thing applies to 月曜, but the difference is that the 曜 feature is only considered attributable to days, so it's almost unable to show a clear "English language muffin" example for this.
That said, you can see a few noninterchangeable circumstances: 暗黒の月曜日 ("Black Monday") wouldn't be *暗黒の月曜, because the exact day is memorable, rather than the the fact the day is accidentally a Monday. On the other hand, I'd say 月曜研究会 ("Monday study group") but not *月曜日研究会, because it'll sound like we're studying about rather than on Mondays. Generally, 月曜 is preferred to make of compound words, since what day it belongs matters more than the real date, unless the very day(s) of certain 曜日 is really in concern (where It's difficult to explain it using English because it lacks a word means "Monday-ness" without saying "-day", but I hope you get it).