From my understanding, all NP-complete problems are NP-hard but some NP-hard problems are known not to be NP-complete, and NP-hard problems are at least as hard as NP-complete problems.
Is that mean NP-hard problems that are not NP-complete are harder? And how it is harder?
From http://en.wikipedia.org/wiki/NP-hard#Examples
There are also decision problems that are NP-hard but not NP-complete, for example the halting problem. This is the problem which asks "given a program and its input, will it run forever?" That's a yes/no question, so this is a decision problem. It is easy to prove that the halting problem is NP-hard but not NP-complete.
The set of NP-hard problems is a superset of the set of NP-complete problems. There are complexity classes more "difficult" than NP, for example PSPACE, EXPTIME or EXPSPACE, and all these contain NP-hard but not NP-complete problems.