Will or Bill?

Why is it if your name is William you then become Bill? Embarassing but true I thought if your name was Bill then your full name was Billiam. I learned when I was a senior in high school that was not the case. I had a friend named Bill and when they called him up for his diploma at graduation they said William *******. My mouth was like what the ....? I thought his name was Bill. That is what I told the gal next to me. Thankfully she was a friend and explained and we had a silly laugh. I just thought it was common sense.
William = Will
Billiam = Bill
that's it and that makes sense!