Adding a suspension to a Bicycle is always a trade-off. You get increased comfort and control, but it adds weight and complexity, and it can make the bicycle pedal less efficiently.
That said, modern suspensions are efficient enough, lightweight enough, and reliable enough that, for many people, the benefits of suspension outweigh the downsides.
The weight and complexity issues are fairly straightforward with more shocks, pivots, and bits of linkage you have on a Bicycle, the harder it will be to keep the weight down. Those pivots and shocks also require regular maintenance to keep everything running smoothly, so that’s just one more thing you have to pay attention to.
The efficiency issue largely has to do with travel in the suspension always working. With most frame designs, force on the drivetrain will have some effect on the suspension. Basically, when you mash on the pedals, part of your power is going into making the bicycle go forward, and part of it goes into making the suspension compress. Generally speaking, this is not a good thing—you want all of your power going into making the bicycle move forward.