Tricky question.
I'll start with Sentence 2:
Nobody ever speaks with me.
This means that nobody speaks with me at any time.
Now, Sentence 1:
Nobody speaks with me.
This means that nobody ever speaks with me.
In short, when the sentences are parsed and analyzed, they pretty much mean the same thing – in the word-by-word literal sense.
However, let's dig a little deeper. Who would ever say such a thing? A monk in a monastary with a vow of silence? (The monk might write it down, but he probably shouldn't say it aloud). A child at school feeling lonely or bullied? That's what I'd be inclined to think.
In that case, is the sentence even true? Surely somebody speaks with the child every now and then. Parents, siblings, teachers, waiters at the restaurant, etc. So, is the child telling a lie? What does it REALLY mean?
I suspect it really means something along these lines:
Hardly anyone ever seems to speak with me, insofar as friends at school go.
I suspect that's the true meaning of the sentence.
If that's the case, I think ever is being used in the sense of Definition #4 at Oxford:
ever (adv.) used for emphasis in questions expressing astonishment or outrage: who ever heard of a grown man being frightened of the dark?
That is, the inclusion of ever is simply a way to make the assertion more emphatic.
One could also argue that the inclusion of words like ever or never also causes a slight shift in emphasis, by introducing an element of time into the assertion:
Jones, no useful findings have come out of your experiments.
Jones, no useful findings have ever come out of your experiments.
Again, there's no reason to think there's any difference in meaning, but the second sentence makes it seem like the experiments have been going on for a long time, and still nothing useful has been found.