Games usually load noticibly faster. When you play a game that has lots of loading screens from changing areas such as Fallout 3 (my favorite 360 game by far), you will notice a big difference. I should also mention that installing games to your HD will effectively prolong your DVD drive's life since it will only be used when you first start the game for a few seconds. You can network your 360 to your PC to stream movie and music files. Divx does work, you can even play avi files from a usb stick put right into the front of the 360. All you have to do is install a small required video update on your 360, which it will prompt you about the first time you try to play a divx movie file. And as far as a sleep mode, the only one I know of is that you can set your 360 to turn off if it's inactive for 6 hours... so ya not that great of a sleep timer but it's something.
Oh, forgot to mention. If you like downloadable content for games you will NEED some kind of HD, the onboard memory on the Arcade is not enough at all. And if you even never go online, there are game packs out there that have downloadable content on a disc that will install onto your HD from that rather than buying it online and downloading it. Fallout 3 Game of the Year Edition is a good example, it has the game on one disc and all of the expansions on the other. You pop the second disc in and you can install whichever of the expansions you want, and then you don't need to use that disc again unless you want to install something again later.
And lastly, there's also free game updates that take up space on your HD. Left 4 Dead and Burnout Paradise had some nice free updates, but if you want them you of course need that HD.