I have a domain for my channel except the website url is actually my name. I'm still tweeking it out and right now it only has the same videos that are hosted on youtube. However, as time goes by there will also be content and interaction that is only available through the website and not youtube, which gives subscribers a reason to follow both resources.
PROS:
Monetization: On youtube I have 1 ad source. On my website I could have a total of 3. Well, the adsense limit is 3. However, I'm still checking to see is the embedded ad in the video actually counts as 1. I'm thinking it doesn't because it's technically not on page scripting. So, you might be able to go 4 ads. I personally only use 2 because I don't want to overkill it during startup. Later it will go to 3 though. Also income from traditional adsense puts youtube to shame. My RPM from those is over $20 and I don't have to share with any network to get it.
Full Control: You don't like the new youtube layout .. your site looks the way it is because you want it to.
SEO: You have full control over the entire spectrum of SEO. Youtube SEO is extremely limited from what I've seen so far. This means you have the chance to attract an audience who .. believe it or not .. never uses youtube.
E-Commerce: If you have physical products or digital downloads you'd like to sell then you can have a shopping cart that can be easily promoted alongside your videos .. in the descriptions .. sidebars .. wherever.
Your Rules: If there's something you would like to do that normally violates the fine print of some youtube policy .. you can self-host the video .. it's your site and you follow your rules. Provided they do not violate the adsense rules .. unless ads are removed from that one page. Not sure what you would do here ... but it would be listed as a pro because it offers more freedom.
Non-Video: You don't always need to produce a video in order to produce content. You could quickly blog something and be done. Which adds more content, more SEO potential, more interaction with followers and so on.
CONS:
Full Control: Something breaks .. you fix it .. or pay somebody to. If you don't know how to build a good site you'll also be paying for help .. or just not look professional.
SEO: You're going to invest time to really study it.
Promotion: You're now spending time promoting 2 sources ... more time investment.
Overall if you're serious .. the pros far outweigh the cons.