Changing site names

[Toucan]

I'm moving a site from an old site name to a new one. The new name has been set up as an alias, but now I want to retire the old name. Can I simply, remove the alias, rename the old site to the new in the control panel and then create the alias out of old name to preserve links?

Also, do search engines view the primary site differently to an alias? Can they even tell the difference?

Many thanks