By default and for security purposes, sudo does not preserve the user environment.
There are multiple ways to deal with this; for your specific case, I recommend adding VISUAL=/usr/bin/vim (and EDITOR=/usr/bin/vim for programs that use the wrong one) to /etc/environment.
Second choice, if you have a permissive sudo configuration (perhaps because it's just you, or all sudoers have real root access anyway) is to run visudo and add Defaults env_keep="VISUAL EDITOR" to allow all sudoers to specify their own (i.e. preserve that environment variable).
Your existing sudoers file may well already have (or include) one or more env_keep statement(s), e.g. this (from the pastebin example):
Defaults env_reset
Defaults env_keep = "COLORS DISPLAY HOSTNAME HISTSIZE INPUTRC KDEDIR LS_COLORS"
Defaults env_keep += "MAIL PS1 PS2 QTDIR USERNAME LANG LC_ADDRESS LC_CTYPE"
Defaults env_keep += "LC_COLLATE LC_IDENTIFICATION LC_MEASUREMENT LC_MESSAGES"
Defaults env_keep += "LC_MONETARY LC_NAME LC_NUMERIC LC_PAPER LC_TELEPHONE"
Defaults env_keep += "LC_TIME LC_ALL LANGUAGE LINGUAS _XKB_CHARSET XAUTHORITY"
The first of those env_keep lines will overwrite whatever edits you make above it (assignment), the other lines extend the list (+=). To keep everything working the way you're used to and unless you have a reason to change the set, I recommend adding a line immediately after those:
Defaults env_keep += "VISUAL EDITOR"
(There are other options to use your preferred editor, but they're either needlessly complicated or overly permissive and dangerous, so you'll have to read man sudoers, figure them out yourself, and understand what the associated problems might be.)
sudo echo $EDITORdoesn't do what you think it does here (and things to try to make your crontab work) – Mat Dec 23 '12 at 12:20