Adding weight is an extremely efficient way to strengthen core, leg and back muscles - and also your fingers, depending on what kind of training you are doing. I have been training with extra weight for quite a while, and I think I can add something to the discussion.
Most people interested in climbing 5.13 will end up finger boarding and campusing. This type of training places a lot of strain on tendons and muscles, being actually not recommended if you're not a regular 5.12 climber. If you are, however, you're bound to climb stuff that requires crimping on small holds. Even the most basic fingerboard training will improve your overall finger strength by an unbelievable amount, but it comes a time when it simply doesn't work anymore. What do you do? Well, you start grabbing smaller and smaller holds, until you injure yourself because you were not strong to be doing that anyway. Besides, that route you want to climb has no tiny holds as the ones you're grabbing, so why train on them? What the route does have is a long sequence of not that bad crimps, from which you must place pro or even rest, and you're absolutely comfortable climbing each individual section, but can't link them all.
The situation above is very common, and a good sign that you should try adding a little weight to your static finger training - this will make your fingers stronger without changing the size of the holds, and therefore will not concentrate strain in a single section of your fingers. It will also make you stronger so that you can start training on smaller holds in the future.
Now, the added weight described earlier is static. Why would I want to climb with weight? Well, suppose your project has a lot of long moves and it's required that you cut you feet lose from time to time. The strength needed to get them back to the rock comes from your core, and training your core by just climbing is very hard - basically because your arms get tired first (even on a roof!). Well, if you tie some weight to your feet, the strength needed to raise them is much, much higher, specially due to increased torque. This will guarantee your core muscles will get beaten earlier than your arm muscles.
Now, if you're interested in endurance training, why not simulate that 11th pitch to the summit, where you're absolutely destroyed and pumped? Climbing a large number of moves with a weight vest will place more strain in your arms, and each move will feel three times as hard. The objective here is to hold big holds (mind that holds that are big for some people might be tiny for you), without placing too much strain on your tendons, but increasing muscle abuse. This is also a good way of simulating the weight of climbing equipment, as you wrote in your question.
As an advice, take weight training carefully. If your interest in it is just about simulating climbing with the equipment and harness, then this is not enough. Weight training should be reserved for more specific problems, like the ones I described above, and should be only pursued by climbers with some experience in grade (above 5.12) and time (2+ years). Also, it should never be done more than twice a week (even most professionals follow this rule), and should be reserved for the middle of your practice: never in the beginning, never in the end.
Also, it is interesting that in you gym you don't usually see people climbing with weight vests: in my gym we usually have to fight to get to the weight closet first!