Your posts are located in the database. If you download all files, you won’t get any posts. You also have to move the database. Export everything from your old hosters database and import it to your new hosters database. Otherwise you will just see the default Hello World.
Just follow the instructions in Moving_Wordpress from the beginning – like esmi said.
They worked for me everytime I moved a blog. (Changing urls, keeping urls, …)
