Both are situational. Most people will say that republic's the greatest, but that aint really so true. If you can identify your victory condition, that will help greatly in your choice. You really only see the benefit of the free unit support for rep with cities size 7+ and later 13+; if you want to be a builder that works out perfectly. You can switch to rep early and stay there the rest of the game, working on space, 20K, or a diplo win. The ancient age republic slingshot is very easy to accomplish at emperor and below, and gives great benefits right away unless you're running around with 8000 immortals or something.
On the other hand, if you are cramped with a coupla other hostile civs on a landmass, and war is needed, its monarchy all the way. Even an oscillating war back and forth would find monarchy as the better choice. I believe Sir Pleb in his "Going for Sid" thread crunched some numbers, compared the two, and found monarchy was his best choice in that game; but again, that was situational. On a higher difficulty level, you usually need to fight to survive, and no matter how many catapults and whatnot you tote around you'll lose units. This = war weariness for your people. Supporting a purely offensive army of say horsemen, and your workers, can add up to a lot in republic, and god help you if for some reason you build spearmen and such everywhere. I think in the end republic needs more close attention/management than monarchy, which many people dont care/dont know how/are too lazy to give.
One thing funny I've noticed about using republic without many luxuries, is that your people arent very happy to start with, you see a new tasty lux right across the border in AI land, decide to go take it, and war weariness sets in bc you are acquiring more luxes for those silly corrupt-floodplain-grow-too-fast-Ijustrushedamarketandyouarestillnothappy clowns. According to the war weariness study in the War Academy (I do not remember the author), each time one of your units is attacked weariness accrues, especially if you are the aggressor. So it becomes a losing gambit if you overestimate how strong you are and how quickly you can take the AI out, and get into a tit for tat skirmish.
In short, republic is the trader/builder/lighter warfare way and monarchy is the heavier warfare option. It sometimes is a tough choice when a game situation falls in between the two types, and thats just a choice you have to make. Though I tend more toward republic, it is a fallacy to blindly go that route without assessing your situation, and too many people follow that path.