$dbHandle->errstr is object-oriented syntax, $DBI::errstr is procedural syntax. In theory, calling errstr using a dbHandle should return the last error on that handle and calling it through the DBI module should return the last error seen anywhere in the database. But, given that

1. Many programs only use a single database connection
2. Many database engines only preserve a single error message per client
3. You should check your error messages immediately, before going on to another database operation

they end up being interchangable in practice.