This past election we heard all kinds of concession speeches and, most frequently, heard the candidates say that they either wished the other candidate well or congratulated them. While both seemingly convey the same message, they are actually quite different.
According to the etymology dictionary, congratulation comes from the latin word congratulationem, which comes from "com" (meaning "together, with") and "gratulari" (meaning give thanks, show joy). When you congratulate someone, you are saying that you are vicariously feeling joy for their accomplishment.
Therefore, there are few occasions when the loser of anything feels vicarious joy for the winner. Congratulations are best suited for weddings, graduations and new parents.
In politics, the only appropriate time to congratulate your opponent is in a primary-- even though you don't mean it, you want to convey the message to your fellow party members that you are happy about the voters choice.
In a general election, however, it is never appropriate to congratulate your opponent. A.) You are obviously not vicariously feeling joy for their victory and B.) The whole basis of why you are running is to deny your opponent and their party a victory and such a sudden aboutface wouldn't make any sense.
So what do you say when you lose? Through clenched teeth, you wish your opponent well. That's a throw-away line that doesn't really mean anything except that you wish him or her well, where "well" is defined by the person saying it.