Move all files except one

A quick way would be to modify the tux filename so that your move command will not match.

For example:

mv Tux.png .Tux.png

mv * ~/somefolder

mv .Tux.png Tux.png

Put the following to your .bashrc

shopt -s extglob

It extends regexes. You can then move all files except one by

mv !(fileOne) ~/path/newFolder

Exceptions in relation to other commands

Note that, in copying directories, the forward-flash cannot be used in the name as noticed in the thread Why extglob except breaking except condition?:

cp -r !(Backups.backupdb) /home/masi/Documents/

so Backups.backupdb/ is wrong here before the negation and I would not use it neither in moving directories because of the risk of using wrongly then globs with other commands and possible other exceptions.

If you use bash and have the extglob shell option set (which is usually the case):

mv ~/Linux/Old/!(Tux.png) ~/Linux/New/

I would go with the traditional find & xargs way:

find ~/Linux/Old -maxdepth 1 -mindepth 1 -not -name Tux.png -print0 | 
    xargs -0 mv -t ~/Linux/New

-maxdepth 1 makes it not search recursively. If you only care about files, you can say -type f. -mindepth 1 makes it not include the ~/Linux/Old path itself into the result. Works with any filenames, including with those that contain embedded newlines.

One comment notes that the mv -t option is a probably GNU extension. For systems that don't have it

find ~/Linux/Old -maxdepth 1 -mindepth 1 -not -name Tux.png \
    -exec mv '{}' ~/Linux/New \;