In C++, there is only a subtle difference. It's a holdover from C, in which it makes a difference.
The C language standard (C89 §3.1.2.3, C99 §6.2.3, and C11 §6.2.3) mandates separate namespaces for different categories of identifiers, including tag identifiers (for struct/union/enum) and ordinary identifiers (for typedef and other identifiers).
If you just said:
struct Foo { ... };
Foo x;
you would get a compiler error, because Foo is only defined in the tag namespace.
You'd have to declare it as:
struct Foo x;
Any time you want to refer to a Foo, you'd always have to call it a struct Foo. This gets annoying fast, so you can add a typedef:
struct Foo { ... };
typedef struct Foo Foo;
Now struct Foo (in the tag namespace) and just plain Foo (in the ordinary identifier namespace) both refer to the same thing, and you can freely declare objects of type Foo without the struct keyword.
The construct:
typedef struct Foo { ... } Foo;
is just an abbreviation for the declaration and typedef.
Finally,
typedef struct { ... } Foo;
declares an anonymous structure and creates a typedef for it. Thus, with this construct, it doesn't have a name in the tag namespace, only a name in the typedef namespace. This means it also cannot be forward-declared. If you want to make a forward declaration, you have to give it a name in the tag namespace.
In C++, all struct/union/enum/class declarations act like they are implicitly typedef'ed, as long as the name is not hidden by another declaration with the same name. See Michael Burr's answer for the full details.