To 'verify' is to check or substantiate the truth of something.
"He verified that the incident actually happened." = "He checked that the incident really did happen"
To 'certify' can mean the above in a more formal sense, but as well it is used to mean that something meets or attains a certain written or accepted level or standard.
"All the parts of a plane are certified to meet a certain standard" = "All the parts of a plane are checked that they meet an approved written standard".