Relic weapons represent the pinnacle of what a player can achieve WITHOUT RAIDING. If you're the type of player who can simply say, "Well, why not just go raid a while and walk away with a better weapon?" then you are not who the relic progression was designed for. Many raiders who strive for relic do it either so that they have a powerful weapon to use as a stepping stone to obtain the ultimate weapon - but that's admittedly a lot of work for only slight improvements over what they could get from lesser raids.
The other reason to obtain a relic weapon is for the sense of achievement (probably the primary reason that a raider would bother with a Relic), and the pride in showing off an item that you worked really hard for. If the extreme grind and huge cost seems annoying and, above all, pointless, then you're probably better off dropping your relic progression and never looking back. Whatever else the relic progression might be, it is a whole lot of work. Folks can argue back and forth all day about the quality of the work (it's not even hard, just a huge grind!) but the fact remains that for many (maybe even MOST) people, it would be LESS work to join a raiding group and get to the point where you have Turn 13 on farm mode. A completed relic represents significant devotion to a task, even if it is a task that you (or even they!) find to be boring, monontonous, and unrewarding.
Well... to be truthful, there IS one other possible reason for sticking with it: Speculation. It's possible that someday in the future SE may "reward" relic users by adding something powerful and/or unique to the weapons, something to make the whole process worthwhile in the end. This would not be the first time SE has done something like this. In FFXI, SE added a bunch of "evolving weapons" through something called the Trials of the Magians. Some of these weapons were particularly grindy, and the end result was nice, but nothing special - at first. As the weapons continued to evolve, though, these grindy weapons got more and more bells and whistles, including extra powerful weaponskills with special buffs tacked on. These eventually came to be known as Empyrean weapons, and were competitive with the most powerful weapons in the game.
Will this ever happen with FFXIV Relics? Will they one day gain something special that propels them into a new league of weaponry? Frankly, I doubt it, but anything's possible.