This is more of a system configuration question, but still …
On the server, everything is locked down based on users and user permissions. You are a user. Your FTP account is a user. Your root account is a user. WordPress itself is a user.
So while you as a user might have write permissions for the WordPress directories (which is why you can manually upload files), WordPress might not have write permissions and won’t be able to add/edit its own files.
There is more information about file permissions located in the Codex, but I highly recommend you ask a system administrator (someone with plenty of Linux experience) to check to see which users have write permission and which user WordPress is running as.