Both are RPM-based and as you are likely to only get shell access anyway both will be as easy to use as each other.
In terms of security, CentOS tends to use older packages which obviously have the benefit of having been out in the wild long enough to have key security holes plugged.
However, the biggest security issues are likely to be in the application-specific configuration of your server processes like Apache, MySQL, SMTP etc where it doesn't really matter which OS you choose.
The downside of CentOS is the packages may be too old for what you need (e.g. I had problems with the version of PHP offered). You can go off-piste and compile/install your own packages but that's an additional layer of hassle and dependency chasing to consider.
The downside of Fedora is a more bleeding-edge approach to packages which may mean less stability long-term.