difference between "function foo() {}" and "foo() {}"

The function keyword was introduced in ksh. The traditional Bourne shell only had the foo () syntax, and POSIX standardizes only the foo () syntax.

In ATT ksh (but not pdksh), there are a few differences between functions defined by function and functions defined with the Bourne/POSIX syntax. In functions defined by function, the typeset keyword declares a local variable: once the function exits, the value of the variable is reset to what it was before entering the function. With the classic syntax, variables have a global scope whether you use typeset or not.

$ ksh -c 'a=global; f () { typeset a=local; }; f; echo $a'
local
$ ksh -c 'a=global; function f { typeset a=local; }; f; echo $a'
global

Another difference in ksh is that functions defined with the function keyword have their own trap context. Traps defined outside the function are ignored while executing the function, and fatal errors inside the function exit only the function and not from the whole script. Also, $0 is the function name in a function defined by function but the script name in a function defined with ().

Pdksh does not emulate ATT ksh. In pdksh, typeset creates locally-scoped variables regardless of the function, and there are no local traps (though using function does make some minor differences — see the man page for details).

Bash and zsh introduced the function keyword for compatibility with ksh. However in these shells function foo { … } and foo () { … } are strictly identical, as is the bash and zsh extension function foo () { … }. The typeset keyword always declares local variables (except with -g of course), and traps are not local (you can get local traps in zsh by setting the local_traps option).

There is no difference AFAIK, other than the fact that the second version is more portable.

foo() any-command

is the Bourne syntax supported by any Bourne-like shell but bash, yash and recent versions of posh (which only support compound commands). (the Bourne shell and AT&T implementations of ksh don't support foo() any-command > redirections unless any-command is a compound command though).

foo() any-compound-command

(examples of compound commands: { cmd; }, for i do echo "$i"; done, (cmd)... the most commonly used being { ...; })

is the POSIX syntax supported by any Bourne-like shell and the one you generally want to use.

function foo { ...; }

is the Korn shell syntax, which predates the Bourne syntax. Only use this one if writing specifically for the AT&T implementation of the Korn shell and need the specific treatment it receives there. That syntax is not POSIX, but is supported by bash, yash and zsh for compatibility with the Korn shell though those shells (and the pdksh-based variants of the Korn shell) don't treat it any different from the standard syntax.

function foo () { ...; }

is the syntax of no shell and should not be used. It only happens to be supported by accident by bash, yash, zsh and the pdksh based variants of the Korn shell. Incidentally, it's also the awk function syntax.

If we continue going down the esoteric list,

function foo() other-compound-command

(like function foo() (subshell) or function foo() for i do; ... done) is even worse. It is supported by bash, yash and zsh, but not ksh, even the pdksh-based variants.

While:

function foo() simple command

is only supported by zsh.