As with ‘there, their and they’re,’ the two words, ‘weather and whether‘ are commonly confused – both spoken and written – words.  These words are homonyms –  which means that they sounding the same, but have different spellings and meanings. In this case weather and whether have very different meanings, and once you know the meanings, you are unlikely to confuse the two again.

weather and whether defined

Weather
Noun
The word to describe atmospheric conditions such as rain, dry, cloudy, plus temperature
The weather was sunny when we began our climb but soon turned overcast and rainy
Weather
Verb
To be (adversely) affected by, worn down or aged by
“The upholstery on the sofa is really weathered.”
Weather is also used as a figure of speech to describe the act of enduring, or getting through something unpleasant.
“They had weathered the tragic events of last summer, and were looking forward to their lives in a new country.”

Whether
This is a conjunction, used when you are talking about possibilities or options.
“I do not know whether he has landed yet or not.”
“Whether we go or not, I am not too bothered.”
Note: If you can replace whether with ‘if,’ then you are using the word correctly.
“If we go or not, I am not too bothered.”

(Digression: There is another word, apart from weather and whether, that is part of this group of homonyms, but it is not commonly used unless you are involved in sheep rearing or related industry.  The word, wether, means castrated sheep.  You are more likely to have heard of the word bellwether, a version of the word wether. Bellwether is indicator of trends, or of change “Kate Moss is a bellwether for high street fashion . The word derives its meaning because a wether, or castrated sheep, is the leading sheep of the flock and wears a bell around its neck to identify it as such.)

As with there, their and they’re, the words weather and whether are pretty commonly spoken and written words.  They are also homonyms – sounding the same, but with very different spellings and, as you will now understand, with very different meanings.