Usando o método self Join().
SELECT a.person,
a.taskid,
TIMEDIFF (DATE_FORMAT(STR_TO_DATE(a.Start_time, '%a, %d %b %Y %H:%i:%s'), '%Y-%m-%d %H:%i:%s') ,DATE_FORMAT(STR_TO_DATE(b.End_time, '%a, %d %b %Y %H:%i:%s'), '%Y-%m-%d %H:%i:%s') ) as Time_between_attempts,
a.Start_time,
b.End_time
FROM test a
LEFT JOIN test b
ON a.person = b.person
AND a.taskid = b.taskid + 1
ORDER BY 1, 2;
Mas isso ignorará o fuso horário.