Weather as a noun is the state of the atmosphere at a certain time and place – sunny, windy, snowing, hot, cold, etc. To weather (the verb) is to change in appearance or form through long exposure to the weather (more commonly: to be weathered).
Under the weather means unwell or depressed – you’re metaphorically under a cloud of unwellness, I suppose!
Whether is a conjunction expressing …
- a choice between two alternatives – “I can’t decide whether to go to the cinema or the park”
- a doubt, either expressing a question or investigation – “I don’t know whether keeping this hat I’ve found is the right thing to do” / “He will check whether she lost a hat”
- an indication that a statement is true, whichever of the alternatives mentioned is the case “I will go to the cinema whether they’re showing a western or a thriller”
Of course, the two can always be found together: “I’ll go out, whether the weather is sunny or wet”, but hopefully you know the difference now!
Edited to add – my friend, Ian Braisby, just reminded me that a wether is a castrated sheep! So now you know …