What is the difference between "gksudo nautilus" and "sudo nautilus"?

Taken from here:

You should never use normal sudo to start graphical applications as root. You should use gksudo (kdesudo on Kubuntu) to run such programs. gksudo sets HOME=/root, and copies .Xauthority to a tmp directory. This prevents files in your home directory becoming owned by root.

Please note that this is primarily about configuration files. If you run Nautilus as root, even with gksu/gksudo, and you create a file or folder anywhere with it (including in your home directory), that file or folder will be owned by root. But if you run Nautilus (or most other graphical applications) as root with sudo, they may save their configuration files in your home directory (rather than root's home directory). Those configuration files may be owned by root and inaccessible when you're not running as root, which can severely mess up your settings, and may even keep some applications from working altogether.

The solution, once you have made this mistake, is to find the configuration files and delete them or chown them back to belonging your non-root user. Many such files start with a . or are contained in a directory that starts with a .. Some are located inside the .config folder in your home directory. To see files and folders that start with a . in Nautilus, press Ctrl+H (this shows hidden files.) To see them with ls, use the -a (or -A) flag.

To find if there are files not owned by you in your home directory, you can use the following command in a terminal:

find $HOME -not -user $USER -exec ls -lad {} \;

which will list all files under the home directory not owned by the user.

Did you know there is a Nautilus add on called nautilus-gksu which adds an "open as administrator" to nautilus' right click menu?

Note: Not available in Ubuntu 12.04 and onward.

If you start a graphical application with sudo you can mess up the ownership of your files which can cause your apps to break. Never do that. Check the Community Help:

You should never use normal sudo to start graphical applications as Root. You should use gksudo (kdesudo on Kubuntu) to run such programs. gksudo sets HOME=~root, and copies .Xauthority to a tmp directory. This prevents files in your home directory becoming owned by Root. (AFAICT, this is all that's special about the environment of the started process with gksudo vs. sudo).