Easy - KKW Standard, or KKW Formal is the Martial Art side of Tae Kwon Do.
The WTF is the Sport Side.
Some Dojangs teach people Tae Kwon Do with an Olympic Mindset.
Some Dojangs teach people Tae Kwon Do with a Martial Mindset, with the Tournaments as just something tacked on.
The easiest way to tell if a Dojang is one or the other, is how many Non-Kicks it uses. Since a WTF Themed Dojang has little use for things that dont tend to score as much, whereas a KKW Dojang will EMPHASISE Kicks, but it still teaches Punches, Elbows, Knees, Grapples, Throws, Takedowns, and some Submissions, as well as Self Defense.
Some Dojangs teach it both ways, which i consider better for a KKW Themed Dojang.
Depicted more easily:
Lets drag up some Kwonkicker videos, shall we

Sport Variations;
Formal Variations;
(Theres a bit of Boxing in the second one, but its mostly KKW Formal)
The main difference is, that one is Technique and Speed (Sport), the other is Technique and Speed and Power and Then Some More Power (Formal)
EDIT: Also, Knife Hands, Palm Heels, Ridge Hands, Backfists, and other such things ought be taught for Formal or Standard, if im not mistaken.