"Se me acercaron" vs "Se acercaron a mí."

"Se me acercaron" vs "Se acercaron a mí."
 

He visto los dos ejemplos usados y no estoy seguro de que sea la diferencia principal entre ellos.

The meaning is exactly the same.
In the first example, you've employed an indirect object pronoun; in the second, an indirect object.
You can also use both in the same sentence.
Se me acercaron a mí.
It may seem redundant to us English speakers to have both an indirect object pronoun and an indirect object, both in the same person, but it is very common usage among Spanish speakers.
There's nothing wrong with any one of these sentences.
The last way I mentioned is also a way to add emphasis.

Another question. Since the indirect object here seems to indicate a destination, can you use the ID with other verbs?

"Venme" doesn't sound right to me.

To approach someone. Acercarsele a alguien.
To come to someone. Venir a alguien
To go to someone. Ir a alguien.

The last 2 examples don't seem right to me.

Verbs that indicate movement are followed by the preposition 'a'.
That explains what is wrong with your first example.

There's nothing wrong with your last two examples.
The first of the three in that group needs an accent mark, but it otherwise fine.

Ven a mí
Acercársele a alguien

Yes, but if I say "venme" that can't be right, can it? But if I say, acercatele, that's ok.

@Babymetal: That's because "venir" is an intransitive verb and "acercar" is a transitive verb. While I can't perform the action of the verb "venir" on any object, I can actually make something/someone "acercar" something/someone else.

Same with verbs like "entrar", "salir", "existir", etc. Although you can put together something like "exísteme", the expression makes no sense, because you can't order anyone to exist on someone.

There is a difference with imperatives like "súbeme", "bájame", "llévame", which make sense because I can actually perform an action to shift a person from one place to another.

I had an idea it was due to the kind of verb it was. Thank you very much. :)