Comparatives and superlatives ("better", "soonest") are adjectives that would not normally take the definite article "the". However, it is grammatical to use the adjective as a noun, where the adjective stands as an ellipsis of a noun phrase. For example:
Which of these shirts do you want?
I'll take the blue (shirt).
How many are in your group?
We are three (people).
When forced to choose between evils, try to pick the lesser (evil).
Among the runners, she is the best (of all the runners).
As in your example, the adjective can also stand for an adverbial phrase that would otherwise include a noun:
We like this book the best (of all the books).
There are also set idiomatic phrases like "the X the better", but again these are elliptical phrases that omit redundant information.
When should I arrive?
The sooner (moment in time that you can arrive), (it will be) the better (situation).
I don't think there is any special rule around the use of this other than the sentence should still make sense with the noun omitted. For example:
When ordering wine at a restaurant you should always pick the least.
The least what? It's not clear from context if I mean "the least expensive wine" or "the least fruity wine" or something else. Of course people say things like this all the time, which can require the listener to ask follow-up questions.
There are other cases where the definite article can be used before a comparative or superlative adjective, for example if the adjective has been converted into an ordinary or proper noun, as in someone's name or title. For example, the boxer Muhammad Ali was nicknamed "The Greatest".
More information.
"Few" and "a few" are significantly different in grammar and meaning.
"A few" is neutral: it simply refers to a small number of whatever, without suggesting any expectations.
"Few" triggers negative polarity expressions (such as "any" and "yet"), and emphasises the smallness of the number, and suggests that a larger number might have been expected.
A few people have registered.
A few people have an idea what's going on.
Contrast this with:
Few people have registered yet.
Few people have any idea what's going on.
both of which are emphasising that the number of people is small.
A few with negative polarity items is not grammatical:
*A few people have registered yet.
*A few people have any idea what's going on.
There is no real answer to your question: like most "why" or "what is the reason" questions about language, the only answer is "because that's the way it is".
Best Answer
Nouns when used to talk about types of things rather than actual instances don't take articles.
So this ...
is fine.
This ...
implies there is more than one "weather" and you've been talking about at least two "weathers" - the phrase "that is a few degrees warmer" further qualifies which "weather."
If you weren't doing something like comparing multiple locations' weather on a map, go with "I would greet weather ..."