Consider:
I shall not go to Paris, nor to Rome.
and
I shall not go to Paris and I shall not go to Rome.

Do they mean the same? (They seem to me to do so.) Is one better than the other? The first looks... um... neater. Also, does the first need its comma?

I can't go on, I'll go on.
Consider: =A0 =A0 I shall not go to Paris, nor to Rome. and =A0 =A0 I shall not go to Paris and I shall not go to Rome. Do they mean the same?

Yes, but I like better: I shall not go to Paris or Rome.

GFH
Consider: I shall not go to Paris, nor to Rome. ... shall not go to Rome. Do they mean the same?

Yes, but I like better: I shall not go to Paris or Rome.

Or "I shall go to neither Paris nor Rome".

Cheers, Harvey
CanEng and BrEng, indiscriminately mixed
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I shall not go to Paris, nor to Rome. and I shall not go to Paris and I shall not go to Rome. Do they mean the same? . . .

Yes. There are simple logical transformations
(not A) and (not B) = not (A or B)
and vice-versa.
(Also: (not A) or (not B) = not (A and B) and vice-versa.)
Is one better than the other? . . .

"Better" depends on context. In the abstract, if we can rationally talk of a sentence so, the first is a bit nicer because more concise.
Also, does the first need its comma?

Yes. The sentence is effectively two clauses with some elision:

I shall not go to Paris, nor (shall I go) to Rome.

A comma is wanted before a conjunction joining two independent clauses.

Cordially,
Eric Walker, Owlcroft House
http://owlcroft.com/english /
Consider: I shall not go to Paris, nor to Rome. and I shall not go to Paris and I shall ... The first looks... um... neater. Also, does the first need its comma? I can't go on, I'll go on.

"I shall not go to Paris, nor to Rome" is considered proper form in the English language. "I shall not go to Paris and I shall not go to Rome" is also considered proper. I.M.O., it is better to phrase the sentence as, "I shall not go to Paris, nor shall I go to Rome." Otherwise, the sentence could be read, "I shall not go to Paris or Rome."