Well, the obvious difference is demonstrated in your main:
struct foo a;
bar b;
baz c;
The first declaration is of an un-typedefed struct and needs the struct keyword to use. The second is of a typedefed anonymous struct, and so we use the typedef name. The third combines both the first and the second: your example uses baz (which is conveniently short) but could just as easily use struct _baz to the same effect.
Update: larsmans’ answer mentions a more common case where you have to use at least struct x { } to make a linked list. The second case wouldn’t be possible here (unless you abandon sanity and use a void * instead) because the struct is anonymous, and the typedef doesn’t happen until the struct is defined, giving you no way to make a (type-safe) pointer to the struct type itself. The first version works fine for this use, but the third is generally preferred in my experience. Give him some rep for that.
A more subtle difference is in namespace placement. In C, struct tags are placed in a separate namespace from other names, but typedef names aren’t. So the following is legal:
struct test {
// contents
};
struct test *test() {
// contents
}
But the following is not, because it would be ambiguous what the name test is:
typedef struct {
// contents
} test;
test *test() {
// contents
}
typedef makes the name shorter (always a plus), but it puts it in the same namespace as your variables and functions. Usually this isn’t an issue, but it is a subtle difference beyond the simple shortening.