My recomendation is to go to u tube. Great info for the most part. Guy on there mhp gardener is one I like.

As for dis advantages, it can be verry expensive unless you are handy and don't mind researching your options.

While I don't have anything against the big companies making money, remember this is as simple as nutrient dissolved in water that a plant grows in with an air gap. That may sound over simplified but meet those requirements and you will get growth.

I would like to do what you are thinking mainly because the rest of the property receives muted light due to all the large, tall trees on southern bordering neighbor's property, but some of the problems I can't overcome so far are:

pitched roof, dangerous, and if something fell especially would cause damage. If you use hydroponics you likely will have a reservoir at or near roof level, so you need a place to secure it. A flat concrete roof is no problem, but a weak wooden roof is, especially with the tendency to promote rot below.

evaporation can be much quicker in my climate from a hydroponic setup, and in summer this will require close management of nutrient strength. It is the same as summer in general, for example, a half gallon per tomato plant per day almost whether the plant is in the rig or not. If you have a pipe to deliver water, it may work, but you if there is lots of evaporation, a recirculating system with a high loss rate will prevent me from simply replacing lost water in many designs with tap, since the tap is somewhat hard and the dissolved solids will accumulate. On the roof I would need to either deliver rainwater collected below, or collect it right up there. There is no place I can store all that much water on my roof.

The climate you live in is really the first thing you should mention because all the problems I have are different from someone in Seattle. Everything probably can be solved, but as Rio mentioned, hydroponics can become costly to build and operate, in my area lightproofing is a big deal since algae gets out of control at the drop of a hat. And to clean and disassemble a hydroponics rig, I'd need to have a smart design, or need to bring it down and this would be a major inconvenience. If you have a different set of problems, they probably all have solutions too, but think a little about the convenience factor, since one of the pleasures of gardening is enjoying the plants growing, but not becoming a servant to keep it going - something too often that happens when you do something out of season. Wind can be more a problem, and without dirt as a heat sink, temperatures on a roof can fluctuate greatly, which is usually not healthy for the plants.

There are some beautiful rooftop gardens that are ecologically friendly to the electric bill below, but I would just be sure that the simple pleasure of checking on the plants was easy enough that it didn't become a hassle, especially after a thunderstorm. For my case, I've considered a drain to waste system where the excess is collected below and can be diluted and sent back up when I check it. I need a strong pump for that. I also can't put shade cloth over the plants on my roof, so if you might need that think of if and how you'd get it done.

If you are really committed, no doubt you can put something really cool together, I would just keep an eye more on the ongoing maintenance than the initial work to set it up. Finally, a garden on my roof that is not enclosed would attract squirrels which would go right through the roof into the attic and nest, as well as raid the garden. While this is more a general problem, it goes into that category of pleasurable gardening, because truthfully I can't do a good job against the pest squirrels, caterpillars, etc. when it is an easy walk, and it is something I do every day and magnified by the added fragility a hydroponic system has compared to some container systems. Ask yourself - how long will I leave this unattended, and what will happen as it is unattended - can I get away for a weekend and if it rains will all be ok?

The next house I live in I hope I can desigh the roof with the idea of having a garden on it in mind ;-)

Don't overload your roofs. They're not generally designed to carry a lot of extra load. With that being said, don't use Ebb n Flo or any kind of bucket system. I would recommend NFT only.
OF course one small DWC system won't damage the roof. Just don't try to feed a family off it.

I directed the op over here. I would like to note that he stated on the other forum that he is an engineer and so structure for the roof and loading aren't going to be a consideration.
It's a roof deck.

The hydroponic system also doesn't have to be recirculating. A to waste system might just be the solution he needs.