It looks as though you are not running the vsftpd daemon Your result
Username 6179 0.0 0.1 2312 234 pts/0 S 00:43 grep ftp
Shows everything running with the letters ftp in the file name
grep ftp is the search itself
/usr/sbin/vsftp is, I presume the client running to access the other box
/etc/vsftp/vsftpd.conf is the configuration file for the server. I guess it was open in an editor?
Whatever the last two are they are not the daemon (server means the same thing)
So fire up the server. To run it once you just enter vsftpd as a command. To run on startup and at all the time you need to add it to your services.
To add vsftpd to the startup list, execute:
# chkconfig –add vsftpd
To list the services that are configured to startup at run-level 3 (Redhat’s default run-level), execute the following as root:
# chkconfig –list |grep “3
Don't worry, you are going about it the right way. If this does not fix it come back. If I cannot get it working someone else will.