Nope. No difference, except that you can enclose the other inside the ones that you are using.
-- No difference between these
myStr = "Hi!!"
myStr="Hi!!"
myStr = [[Hi!!]] -- The 'weird' way to make a string literal IMO...
-- Double quotes enclosed in single quotes
myStr="My friend said: "Hi!!""
-- Single quotes enclosed in double quotes
myStr = "My friend said: 'Hi!!'"
-- Single and double quotes enclosed in double brackets
myStr = [[My friend said: "What's up?"]]
See: Strings in Lua for more information.