Bitstamp has much higher volume than Coinbase. If you are trading big amounts of Bitcoin it's easier to get your orders filled fast at Bitstamp. Although Bitstamp requires KYC documents to register it's less intrusive than Coinbase. Coinbase also requires KYC documents to register, but it wants to know where your Bitcoins have come from, and where they are going to. You cannot sell Bitcoins you buy there on Localbitcoins, or vice versa. It sometimes closes people's accounts for sending coins to Localbitcoins.
However, Coinbase is insured as this statement from their website shows.
Coinbase is insured against employee theft and hacking in an amount that exceeds the average value of online bitcoin
Up until last month, I had used Coinbase from it's beginning. They have made great strides and caused a few irritations with the in-flux rules they would change weekly. They closed my account without warning after me notating on a few of my transfers that I was sending coins to my BTM wallet. I don't really fault them for wanting to limit those transactions, but would have preferred a little note not to do it anymore. Instead, my account was closed and from my email conversations with them, they won't even let me open a personal account with them anymore. All of that over < 10-15btc over 5 months.
I'm still looking for a good USA exchange I can use to fund my vending machine.