Yes they are different. I'll show you an example. I don't like seafood, but I am willing to try it. Willing is more of a desire to do something. Also, when you make an appointment, the secretary will ask you "when are you available"? Meaning what time can you be there. or I am willing to try it tomorrow, but I'm not available to do it today.

No, they are not the same. Check a dictionary for their meanings.