What are the advantages and disadvantages of keeping bitcoins in an exchange versus a private wallet? What are the advantages and disadvantages of keeping bitcoins in an exchange versus a private wallet?

Exchange advantages: 


Liquidity - you can quickly trade bitcoins for dollars, and vice versa.
Can hold dollars directly


Wallet advantages:


No reporting requirements
Can quickly get at your bitcoins
From what I understand it comes down to how you feel about security, to have a wallet stored privately means that you run the risk of equipment damage where you'll have to spend time on the recovery.

On the other hand you have the risk on network security, at least 2 exchanges have had high profile thefts occur in past years which have effected the market, exchanges make for big money targets and there's no real way to recover the funds.

Personally I think there is likely room for a hybrid solution between the two but I haven't seen anything yet.