Curl uses external library and it has a lot more power to customizing the request - custom headers, generating POST request, uploading files. Everything you need I must say.
Fopen is limited to only just make a GET request of the url without any further customization.
As for security CURL is not influenced by security configuration in PHP (like forbidden fopen of remote URLS and such).
The both possibilities return you data which you can use in every possible way you want. If you make a security hole in your project, after getting the data is your fault after all.
Also I am not quite sure but I think that fopen cannot handle SSL (https) connections. Both fopen and CURL support SSL (as noted by Andy Shellam in a comment below).