I currently have a single "storage account" on Windows Azure for application data. Note that this storage account is tied to the CDN. I want to start collecting diagnostic data and persisting it to a storage account. Should I set-up a new storage account for this or is it ok to use the existing account?
You should use a new account. Definitely recommended that you use a different account in fact (I build a product that monitors Windows Azure diagnostics). Remember that there is an overall Tx limit to your storage account (5000 tx/sec). You do not want diagnostics impacting the performance of your storage account. There are other reasons as well: for instance, if you outsource your monitoring it is nice to be able to give the entire account over for inspection without exposing your application data. Deletion of data and account rolling is much easier as well (when trimming old data).
I think David recommends it too (see #1). That, by itself is reason enough. ;)