If you use any HD wallet on your computer, no one besides you knows that these addresses belong to the same person or even belong to the same wallet. Spying on your computer and analysis on the blockchain being excluded, of course.

Therefore, these transactions go through just like you were sending money to someone else.

If you use an online service, the service provider knows that these addresses belong to the same account. What they do in such a case is any service's decision but most will probably treat it like any other transaction. blockchain.info seems to be no exception: https://bitcointalk.org/index.php?topic=95809.0

Everyone else will just see that money has been moved from one address to another.

This, of course, doesn't always apply. There are exchanges (like bitcoin.de) which allow transfer from the exchange to an address chosen by the user but which are not meant for normal payments. They don't store the funds in addresses specific to a user. When funds are moved away from the exchange, the requests from different users to do so are accumulated for some time (which is why it takes more time to move funds away from such an exchange than would be desirable for payments) and then processed in a single transaction. Such services likely won't transfer any money through the blockchain if you make a transfer from your account to your account's deposit address.