As a noun, weather refers to the conditions outdoors. It includes heat, dryness, sun, wind and rain. Weather as a verb can also mean to successfully come through a difficult situation (like a storm) without being damaged too much.
Examples:
1. I checked the weather for this week because I want to go out.
2. Jack love winter weather because they can go skiing.

whether is a conjunction. In grammar, conjunctions are words that are used to connect two sentences or words within the same sentence. Words like and, if, or, and but are examples of conjunctions.Weather is used to show two alternatives.
Examples:
1. I didn't know whether I should stay or go.
2. It's hard to tell whether the team will win or lose.