A Script that checks if the values are equal?

Firstly, let's clean up that code of yours. Clean code is essential to readability.

-- Value Locate
local loader_Version = game.ReplicatedStorage.Versions --Example: Value is 1
local Module_Version = game.ReplicatedStorage.Offical_Version --Example: Value is 2

-- Updates Tab
local Updated = "Up-To-Date ?" -- Reccomended to keep variables local
local Update_Required = "Loader! ?"


while true do
    if loader_Version.Value == not Module_Version.Value then
        script.Parent.Text = "Update: " .. Update_Required  --When both value are not equal
    end
    if loader_Version.Value == 2 and Module_Version.Value == 2 then do
        script.Parent.Text = "Update: " .. Updated
    end
    wait()
end

Now, we can easily see the errors:

1. On line 11, you're using the not operator, which isn't ideal in this case. To check for inequality, use ~= or the less than (<) operator in this case. Here you can learn more about the not operator.
2. Following the first statement, you can use else to determine what happens otherwise. An if statement works as follows:

if (condition == true) then
... do stuff ...
else
... do something else, as condition is falsey ...
end

Meaning we could do something like this:

if Current_Version < Latest_Version then
    print("I need an update!")
else
    print("I am up-to-date!")
end

In plain English, this is checking if Current_Version (the current version, respectively) is less than the Latest_Version. If so, the program's version is less then the latest version, therefore needing an update. Elsewise, the program is fine.
Here's what it'd look like in your code:

-- Value Locate
local loader_Version = game.ReplicatedStorage.Versions --Example: Value is 1
local Module_Version = game.ReplicatedStorage.Offical_Version --Example: Value is 2

-- Updates Tab
local Updated = "Up-To-Date ?"
local Update_Required = "Loader! ?"


while true do
    if loader_Version.Value < Module_Version.Value then
        script.Parent.Text = "Update: " .. Update_Required
    else
        script.Parent.Text = "Update: " .. Updated
    end
    wait()
end

If I helped, be sure to upvote/accept!

Fixing your script, it would be:

--Value Locate
local loader_Version = game.ReplicatedStorage.Versions --Example: Value is 1
local Module_Version = game.ReplicatedStorage.Offical_Version --Example: Value is 2

---[[Updates Tab]]--
Updated = "Up-To-Date ?"
Update_Required = "Loader! ?"

while wait() do
    if loader_Version.Value ~= Module_Version.Value then -- "do" shouldn't be put after an if-then statement
        -- inequality is tested for using "~=", not using "== not"
        script.Parent.Text = "Update: " .. Update_Required  --When both value are not equal
    end -- you had too many "end"s here
    if loader_Version.Value == 2 and Module_Version.Value == 2 then
        script.Parent.Text = "Update: " .. Updated
    end
end

However, it's better to change the text only when the version updates instead of in a loop to save CPU cycles:

--Value Locate
local loader_Version = game.ReplicatedStorage.Versions --Example: Value is 1
local Module_Version = game.ReplicatedStorage.Offical_Version --Example: Value is 2

---[[Updates Tab]]--
Updated = "Up-To-Date ?"
Update_Required = "Loader! ?"

function checkUpdate() -- put everything in a function
    if loader_Version.Value ~= Module_Version.Value then -- "do" shouldn't be put after an if-then statement
        -- inequality is tested for using "~=", not using "== not"
        script.Parent.Text = "Update: " .. Update_Required  --When both value are not equal
    end -- you had too many "end"s here
    if loader_Version.Value == 2 and Module_Version.Value == 2 then
        script.Parent.Text = "Update: " .. Updated
    end
end

checkUpdate() -- check update in the beginning

function valueChange(property)
    if property == "Value" then -- if it's the "Value" property that changed:
        checkUpdate() -- check update
    end
end

loader_Version.Changed:connect(valueChange) -- when loader_Version changes
Module_Version.Changed:connect(valueChange) -- when Module_Version changes