What are the advantages of that against running your bitcoind on your own server?
Like most things it's a trade off, you need to trust blockchain to not feed you bogus data and you need to have your use case covered by their API.
Oh, and like any service they sometimes have outages, but on the other hand maybe you'll have outages of your own bitcoind.
If you can avoid running the client locally I'd advise to use the blockchain API.