$DBI::errstr is actually a variable, not a method
call. It holds the text of the last error on
any of the database handles.
$dbh->errstr is a method call on a database
handle. It returns the text of the last error
from "the last driver function called," according
to the DBI manpage.
In general, you should use $dbh->errstr, unless
you don't have an available database handle. For
example, if you're calling DBI->connect, you're
creating a database handle, so you don't actually
have one to call the errstr method on. So if
you're doing error checking on the connect
method, you should print out $DBI::errstr if
you get undefined back from connect.
When you're doing error checking for any database
handle methods (prepare, execute, etc.), you
should use $dbh->errstr. (Or just set RaiseError.)