The nice thing about the new homepage is that you can click an option from the left sidebar and the corresponding videos are displayed without opening a new page. Now you can quickly access the list of videos you liked or your history, the latest videos from a channel and the most popular YouTube videos directly from the homepage. Unfortunately, the updated YouTube.com doesn't look great when you're not signed in since most features are missing and YouTube emphasizes the "sign in" button.
YouTube also updated video pages. The background is now light gray, just like on the homepage. Buttons are embossed, there are new icons for the buttons and for the full screen/larger player options.
To try the redesigned homepage and the updated interface of the video pages, edit the VISITOR_INFO1_LIVE cookie for youtube.com and use the following value: ST1Ti53r4fU. For example, in Google Chrome, go to YouTube's homepage, open the JavaScript console (Ctrl+Shift+J or Menu > Tools > JavaScript Console) and paste the following code:
document.cookie="VISITOR_INFO1_LIVE=ST1Ti53r4fU";
Then press Enter, close the console and reload YouTube's homepage.
You can also use extensions like Edit this cookie (Chrome) or Cookies Manager (Firefox). Opera's cookie manager is really advanced, so you don't need an extension to edit cookies.
To go back to the old interface, open your browser's cookie manager, search for youtube.com and delete the VISITOR_INFO1_LIVE cookie.
{ Thanks, Emerson and Mushaf.}