I've read a few answers on here that say when your mysql server won't start in MAMP you should run
sudo killall -9 mysqld
[04-May-2014 11:02:08 UTC] PHP Warning: mysql_connect(): Can't connect to local MySQL server through socket '/Applications/MAMP/tmp/mysql/mysql.sock' (2) in /Users/Chappell/Websites/LFB/wp-includes/wp-db.php on line 1372
The problem was quite simple in the end. I was using different versions of MAMP on the two separate computers. I just installed the latest version and it's working like a charm.