. . . What would you say were the advantages of running the local client over anything web based? . . .
When running the Bitcoin-Qt local client you are contributing to the robustness of the bitcoin network because your client is one more node validating and relaying transactions. When you run a web wallet you are not contributing in this way.
Also, when using a web wallet, there is a slight possibility that a hacker could see the web wallet provider as a high value target since they handle transactions for so many people. With blockchain.info, you run the risk that a hacker could take over their site and put up fake web pages that trick you into providing them access to all the coins in your wallet. You also run the risk that the owners/operators of blockchain.info suddenly decide to try and steal bitcoins from their users and put up "updated" web pages that allow them to gain access to your coins and steal them.
There are browser plug-ins that validate the content of blockchain.info's web pages to make sure that your aren't using fake page designed to steal your bitcoins. If you are going to maintain a significant amount of bitcoins there, I'd suggest using their wallet backup service (in case they shut down) and the browser plug-in (to protect your coins from theft).
. . . Edit: I opened an account at bitchain.info but it's got a different address than the one that's set up in my bitcoin-qt client.
I would prefer to have just one account. But there was no way of changing it on bitchain.info, that I could see.
I thought blockchain.info provided a way to import a Bitcoin-Qt wallet. However, if you've decided to abandon the Bitcoin-Qt client completely, and haven't had any bitcoins sent to that address yet, then you don't need to worry about that address. You can just abandon it and use the blockchain.info wallet instead.
If you are planning on temporarily using the blockchain.info wallet, with the possibility of using the Bitcoin-Qt wallet later, then I'd keep the 2 separate. If/when you decide to switch to the Bitcoin-Qt wallet, you can always send the bitcoins that you have at your blockchain.info wallet to the address of your Bitcoin-Qt wallet and then abandon your blockchain.info wallet.
It is a "bad idea" to try to run both a blockchain.info wallet and a Bitcoin-Qt wallet with the same bitcoin addresses in both. The only reason I'd suggest importing your Bitcoin-Qt wallet into your new blockchain.info wallet would be if you have already given out your Bitcoin-Qt address to someone and are expecting coins to be sent to that address. That way those coins would show up in your blockchain.info wallet.
Note: For the purposes of increased anonymity it is generally recommended that you use a different bitcoin address every time you receive bitcoins. If you follow this advice you'll get used to having more than one bitcoin address no matter what wallet you use.