Rainer's answer discusses the things that you can do with function that you can't do with symbol-function, namely retrieve the value of lexically scoped functions, but there are a few other differences too.
The special operator FUNCTION
The special operator FUNCTION provides a way to find the functional value of a name in a lexical environment. Its argument is either a function name or a lambda expression. That function can take a lambda expression means that you can write: (function (lambda (x) (list x x))). The term function name includes more than just symbols. It also includes lists of the form (setf name), which means that you can do things like (function (setf car)).
The accessor SYMBOL-FUNCTION
The accessor symbol-function, on the other hand, lets you retrieve and set the functional value of a symbol. Since it requires a symbol, you can't do (symbol-function (lambda …)) or (function (setf name)). Symbol-function also can't see lexical environments; it only works with global definitions. E.g.,
(flet ((foo () 'result))
(symbol-function 'foo))
;=> NIL
Since symbol-function is an accessor, you can change the value of a function's symbol with it. E.g.:
CL-USER> (setf (symbol-function 'foo) (lambda () 42))
#<FUNCTION (LAMBDA ()) {1005D29AAB}>
CL-USER> (foo)
42
CL-USER> (setf (symbol-function 'foo) (lambda () 26))
#<FUNCTION (LAMBDA ()) {1005D75B9B}>
CL-USER> (foo)
26
The accessor FDEFINITION
There's also the accessor fdefinition which is kind of like symbol-function in that is can only access globally defined functions, but kind of like function in that it can access of symbols and (setf name) lists. However, it does not retrieve the value of lambda functions.
function
(#'
) is largely equivalent toquote
('
) and hence it does not return the function definition for a symbol; it just returns the symbol. There are still reasons to use it (historically the benefit was restricted to byte-compilation, and nowadays there are also reasons related to lexical binding), but your specific question title makes very little sense for elisp. – phils