Depending on what kind of limited slip, the only real drawback is cost. The Auburn-style cone limited slip like GM used for many years is nice, but not rebuildable. The Eaton-style clutch posi is rebuildable and offers a touch better adjustable torque bias. The Gov-lock is basically a clutch-style posi that is open until wheel slip is detected, then it applies pressure to the clutches and it operates much like an Eaton posi.
There are countless kinds, so I won't go into them all, so it might help if you just tell us what the application is. That way we can tell you some pros and cons of the few posi's that would be effective in your application.
Depending on the type of posi there are huge benefits and some drawbacks, but an open differential's only benefit (in my opinion) is cost. If you're willing to pay for the right posi, you still get the transparent operation like an open diff, but when you need the traction its there at both wheels.