There is no hosted solution for vBulletin (at least offered via jelsoft). But that dose not mean you have to know how to set up and run a server to use vBulletin.
Unless you are going with a un-managed dedcated box or co-location your webhost will take care of the day to day up keep of the computer you are hosted on.
The one thing vBulletin will always hold over ezboard (and other hosted solutions) is that you control how your data is stored, and presented. If you ever need to make modifications to the software to suit your own purposes its possible because you can edit the source code or have someone else do it for you. With ezboard you have to take what they will give you.
Secondly vBulletin will be on space/bandwidth you control at least to a certain point, meaning you don't have to host ad's for others on your board like you have to at ezboard (last time I looked their free acounts still required pop-ups/banners). And if you did want to serve ad's you would get the profit, rather then the company hosting the board for you.
If you are worried about having to install vBulletin yourself jelsoft will do it for you for $135. However I do advise attempting to install it yourself, as it's very easy to do. Upgrades are also a simple process that dos'nt take more then a few minutes (less if your blessed with broadband

).
I encourge you to try the
Admin Control Panel for yourself. This will give you an idea of exactly what vBulletin will and will not do 'out of the box'.