Actually work vs Actually does work?

They technically mean the same thing (and both are grammatically correct).

There is a difference, but it is subtle. It's more clear in spoken English, or at least easier if you add an emphasis where necessary (which I have done below).

It works.

Plain and simple, the thing works; no implications here.

It actually works.

Think of it more as "You would think it wouldn't work, but it works." Perhaps other products in the same category don't work, and this dispels any disbelief in the product's ability to work.

Alan: This application does not work!
Bob: It actually does work, but you need the latest JVM installed.

Even more emphasis on the fact that you think it wouldn't work, but it does.

And if you want to add even more emphasis (but is on the brink of grammatically incorrect, and overuse of this might lead people to stop taking you seriously):

It really does actually work.

The first sentence is just a normal sentence. The second case is a use of to do as an auxiliary verb for emotive emphasis, in an affirmative sentence.

We do not normally use do or does in affirmative sentences, Ali, but we can use them for emotive or contrastive emphasis when we feel strongly about something:

She thinks he doesn't love her, but he does love her. He really does!
You do look pretty in that new outfit! Quite stunning!
Are you all right? You do look a bit pale.
Do please sit down.
I don't see very much of my old friends now, but I do still email them.
Was that a joke? I do believe you're teasing me!

So, the difference between the sentences is the second sentence places emphasis on the verb to work.