I have a strange problem with cloning a git repository from an amazon ec2 server. It works without any problems on one of my computers running ubuntu 12.04, while on another
It usually is a permission issue.
The chmod on the parent directories of your config file might be different between your two computers.
(and I am not talking about just the immediate parent directory .ssh, but also all the parent directories)
See "Git SSH authentication", but also know that if any of the parent directories is writable for group or world, ssh won't work.
Note also that your second command is not right, and should be:
git clone ubuntu@ec2xxx.compute-1.amazonaws.com/var/www/project.git
no ':' (a ':' means using a config file, with an scp-like syntax)
it can only work if you have ~/.ssh/id_rsa and ~/.ssh/id_rsa.pub though.
If you have mykey.pem, then you need a config file for ssh to know where are your public and private keys, which means only this can work:
git clone ec2server:/var/www/project.git
One other chack (after this thread and this forum) is to check if there is any DNS/DHCP issue (a bit like in "Working with git behind a dynamic DNS").
Host ec2server
Hostname 1xx.xxx.xxx.xxx # ip address of ec2XXX.compute-1.amazonaws.com
User ubuntu
IdentityFile ~/.ssh/mykey.pem