1. Automatics still use gears and have to be shifted to specific gears. CVTs don't have gears, they usually (not all, Nissan's Extroid for example) use a pulley system with a metal belt linking the driving and driven pulleys. The pulleys can change their dimensions such that it can vary the overall gear ratio "infinitely" as opposed to having fixed gears. This is supposed to give it fuel efficiency benefits over a traditional geared automatic.
2. The disadvantage with it is depending on the person and specific CVT, it may not simulate the feeling of the shift and the person might be turned off if it doesn't simulate the shift.
3. CVTs are relatively common now but you might have some service issues if you or someone else uses the wrong fluid. Not only do they require their own fluid, sometimes car manufacturers use multiple types of CVT fluids. For example Honda has more than one for different generations of their CVT, Nissan has one fluid for Extroid, another fluid for the other CVTs.
4. Reliability is of concern. Some CVTs are notorious for failure, like the Jatco ones Nissan used in a bunch of Sentra models. Some are better than others here.
5. If you're a tuner, sometimes the CVT limits what you can do to make more power. If the engine makes too much power it can cause severe damage to the belt on some CVTs where as it might not have with a traditional automatic or a manual. In other words the CVT was engineered to perform within a certain range of power and going past it is not an option.
https://www.youtube.com/watch?v=ocR1D_dXhCk
Before I had a CVT, I would have preferred a traditional automatic to a CVT. After actually owning a CVT, I don't mind it any more and would rather have the CVT.