Boy this is hard. I am not the biggest fan of Win 8, but I'll try to be objective.
Does Windows 8 handle media (videos, music, etc) better?
IMO, no. To save some money, Win 8 removed some of the codecs for playing media, specifically DVDs. If you want to play a DVD with Win 8, you'll need a 3rd party app or you'll have to buy the functionality from the Win app store. For people that say DVDs are dead it may not be a big deal, but to answer your question, I believe Win 8 offers less when it comes to media, thus Win 7 is better in this category.
Yes, Win 8 boots up faster when compared to Win 7, but I don't see it being that much of a big deal. I have the BIOS on my main computer set to auto-boot at the same time my alarm goes off (and most motherboards have this functionality). By the time I get to my computer it's up, running and ready to go. If you have an SSD as your boot drive, the difference is 2-3 seconds.
The one thing Win 8 does do appreciably better than Win 7 is handle larger size drives (> 2TB). Win 7 can use 2TB+ drives for storing files, but I've noticed you can't use the built-in Win 7 backup on larger drives. And while this is the issue that impacted me, there can be other circumstances. Win 7's issues stem from the method data is stored (4K v 512 sector size). Long story short, Win 8 is more capable.
Hope this helps.