I think the easiest way would be a SELECT INTO Statement.
http://www.w3schools.com/sql/sql_select_into.asp
Code:
INSERT INTO player_analytics_table1 (`server_ip`, `name`, `auth`, `connect_time`, `connect_date`, `connect_method`, `numplayers`, `map`, `duration`, `flags`, `ip`, `city`, `region`, `country`, `country_code`, `country_code3`, `premium`, `html_motd_disabled`, `os`)
SELECT `server_ip`, `name`, `auth`, `connect_time`, `connect_date`, `connect_method`, `numplayers`, `map`, `duration`, `flags`, `ip`, `city`, `region`, `country`, `country_code`, `country_code3`, `premium`, `html_motd_disabled`, `os`
FROM player_analytics_table2;
It's a select + insert with a new ID
Don't forget to backup first.