No, they don't mean EXACTLY the same thing. The word translated to "whole" is an adjective which modifies a noun or pronoun.
If I say that you are a "male", that is a general basic term that is not descriptive, nor identifies your skin color, age, height, weight nor anything else about you.
When I add an adjective(s) such as "white" or "80 year old", or "6 foot", these terms further describe or modify the word "male".
So when we see in 1 John 2:2 that Jesus is the propitiation for the sins of the whole world, the word "whole" tells us that John means the "whole" (complete, total, entire) world.
But that verse is not speaking of His role as "Savior".
When we see in 1 John 4:14 that Jesus was sent to be the Savior of the world, the word "whole" is not there.
You are beginning to sound like Mr Religion. One big smoke screen. When the scripture says world, it means the whole world. Unless it says some of the world. There is no scripture that says Jesus died for some of the world.