What is the difference between air system and water system in Central air conitioner?

Very broad domain question, in short we can say it depends totally upon the place to be cooled, applications in general, climatic conditions where system will be used. Availability of water. Each factor further has so many details that not feasible to cover here. But one thing is proved that efficiency of DX system is better than chilled water system. Why air is more efficient is simply because of less mediums involved in heat transfer. Difference in initial and running cost varies too.

More specific question will do good than general questions.

- for chilled water system we must have a pipe system and air ducting system.

- if you are asking about the difference between chilled water system and a DX unit connected with air ducting system so it is as following:

- for the same tonnage you will need a bigger duct volume for air system.

- for chilled water system you will need more equipment like chiller, pump, cooling tower

- air system had some limitation for some application like using a DX unit as a fresh air unit.

- for desert country using water system is not good option.

- chilled water system is recommended for high rise building.

the main difference is that you can transfer the same amount of heat by a small amount of fluid volume  in the chilled water system if you compare it with air system

Are you referring to Air cooled and water cooled condensing system in central air conditioner? or if you are talking about air distribution system and chilled water circulation system. The former had a ducting installed and concealed in the ceiling for the  distribution of condition air for each cooling space, where as the latter uses chilled water piping and a fan coil unit strategically installed in cooling space to condition the air.