The course handicaps shown on the My Handicap page are the standard course handicaps from each of the tees at El Tigre. You would use this handicap when all players in a competition are playing from the same tees. So if you and a couple buddies are playing a friendly game, and you all play from the blue tees, you would all use your blue tee course handicap.
The league handicaps are used in any El Tigre league event in which players play from different tees. For instance, in the Mens League, we have players playing from the blue, white, and gold tees. According to USGA/RCGA Handicap Manual paragraph 3-5, course handicaps must be adjusted to account for the difference in course ratings among these different tees.
The El Tigre leagues have standardized on white tees for men and red tees for women, because those are the tees the majority of players use. So your league handicap and course handicap from the white tees (or red tees for women) will always be the same. But if you play from the blue tees, you're playing a longer and more difficult course than the players on the whites, so we add 2 strokes to your blue tee course handicap to arrive at your league handicap. Likewise, if you play from the gold tees, you're playing a shorter and easier course than those on the whites, so we deduct 2 strokes from your gold tee course handicap.
Bottom line: Use the league handicap when playing in one of the El Tigre leagues, because you're playing against players on different tees, and these league handicaps have already been adjusted in accordance with the handicap manual.