What is the difference between a hospital certificate of birth and other birth certificate which is needed to cross Canadian border?

A hospital certificate is a pretty piece of paper. It might even have the baby's footprint on it. It doesn't have very many details. You need the real birth certificate. It has the parents complete names, the date & the time you were born, your complete name. You can get a copy of this from The Bureau of Vital Statistics at the capital of the state you were born in.