purpose of typedef struct etc. ?

[Remington]

I see this kind of thing a lot:

typedef struct
{
//
} name;

or this

typedef enum
{
//
} name;

My question is why do people do this? Does this notation have some kind of special function? Isn't it simpler to write:

struct name
{
//
};

and

enum name
{
//
};

????
Might it have something to do with C vs. C++ if so why do I see this type of notation in purely c++ files which have c++ only code all over the place?
I have never come across an instance where I would even concieve of using the previous method so if somone would be so kind and explain what the point of that is I would be grateful.