A stock exchange is a centralized forum for trading stocks. The advantages of trading such an asset on an organized exchange are numerous:
1) When a stock is traded on an exchange, typically there are rules for allowing that transaction to take place. For example, stocks traded on the NYSE cannot be involved in bankruptcy proceedings. If a stock goes bankrupt, it is delisted from the NYSE. There is more oversight over the transaction so there is less chance for any illicit activities.
2) Also, when stock is traded on an exchange, typically a clearinghouse is used to maintain a record of the transaction. There is a type of risk known as counterparty risk, which is the risk that one party may breach the terms of an agreement. A clearinghouse typically offers insurance against counterparty risk to that possibility is reduced.
Overall, the advantage of having stocks traded on a stock exchange is that regulation is possible since the stock is traded on a centralized platform.