Why is it incorrect to say/write "I want to know today weather" instead of "I want to know today's weather"? [closed]

Well, I think the problem with your reasoning is that "today" is not a standard noun.

We can use standard nouns in this construction, such as in "I want to know the mountain weather" or "I want to know the aviation weather"—both of these sentences are okay (though they sound just a little strange).

If "today" were a standard noun, then we would be able to say "I want to know the today weather". But it's not, so we can't.

Jason Orendorff's answer to “Is ‘yesterday’ a noun, an adjective or an adverb?” states that according to The Cambridge Grammar of the English Language, the word "today" is actually a pronoun, not a noun. And pronouns cannot be used like attributive nouns.