Difference between ~= and the not operator?

Could someone clarify the difference between these two! I’m kind of confused of when these should be used and what is the difference?

the not operator tells you if a value isn't truthy (or is falsy), while the ~= operator tells you if two values aren't equal

in lua, a value is truthy when it isn't nil and isn't false, but in other programming languages, values like 0 and "" are also not truthy

also, not has "operator precedence" over ~=, which means that not is evaluated first

operator precedence is like the programming version of PEMDAS or BIDMAS, and you can find it here https://www.lua.org/pil/3.5.html

local foo = 1
local bar = false
if not foo == bar then -- you might expect this to not print, but because "not foo" is evaluated first, it becomes false, since foo is a truthy value
	print("awesome! not foo equals bar!")
end

sometimes, not and ~= accomplish totally different things

-- this VV

local idiot = game:FindFirstChild("Workspace")
if not idiot then
	print("idiot!") --> doesn't print
else
	print("stupid") --> prints
end

-- is functionally the same as this VV

local idiot = game:FindFirstChild("Workspace")
if (idiot == nil or idiot == false) then
	print("idiot!") --> doesn't print
else
	print("stupid") --> print
end



-- whereas this VV

local idiot = game:FindFirstChild("Workspace")
if idiot then -- putting a value just by itself checks if it is truthy
	print("idiot!") --> prints
else
	print("stupid") --> doesn't print
end

-- is functionally the same as this VV

local idiot = game:FindFirstChild("Workspace")
if (idiot ~= nil and idiot ~= false) then
	print("idiot!") --> prints
else
	print("stupid") --> doesn't print
end