Here is the proof of Gersonides [Levi ben Gershon] (1343) for $2^n-3^m=1$. It uses nothing more that arithmetic modulo $8$.
Case I: $m$ is even. Then $3^m$ is 1 mod 4, so $2^n$ is 2 mod 4, implying $n=1$ and $m=0$.
Case II: $m$ is odd. Then $3^m$ is 3 mod 8, so $2^n$ is 4 mod 8, implying $n=2$ and $m=1$.
The alternative equation $3^m-2^n=1$ follows similarly when $m$ is odd, but is a bit more tricky when $m$ is even (hint, factor $2^n=3^m-1=(3^{m/2}+1)(3^{m/2}-1)$ and argue from there).
Seeing that the link is to a 1996 announcement that I posted to
sci.math.research, I suppose I should explain. Yes, the formulation
in that announcement is not clearly stated $-$ one doesn't polish
USENET posts like a published paper, and can't even edit after the fact
(as is possible on mathoverflow) to correct blatant typos like the stray
"+1" in the formula "(x,y,z)=(1+6t^3+1,1-6t^3,-6t^2)".
As it happens here there is a
published paper
that appeared only a few years later:
Elkies, Noam D.: Rational points near curves and small nonzero
$|x^3-y^2$| via lattice reduction,
Lecture Notes in Computer Science 1838 (proceedings of ANTS-4, 2000;
W.Bosma, ed.), 33-63 (arXiv:math.NT/0005139).
but the relevant section (3.2) doesn't address parametrizations of
$x^3+y^3+z^3=2$. The answer to the present question is that
D.Burde is basically right: the correct statement was, and still is,
that all known solutions of $x^3 + y^3 + z^3 = 2$ in ${\bf Q}[t]$
come from the identity
$$
(1+6t^3)^3 + (1-6t^3)^3 + (-6t^2)^3 = 2
$$
by permuting $x,y,z$ and substituting some polynomial for $t$.
The substitution need not be linear, but nonlinear substitutions like
$(x,y,z) = (1+6t^6, 1-6t^6, -6t^4)$ give no new $(x,y,z)$ solutions either.
I don't think any method is known that would prove that there are
no other nonconstant solutions, or that there's no nonconstant solution in
${\bf Q}[t]$ to $x^3+y^3+z^3=d$ unless $d$ is a cube or twice a cube.
All that can be said is that if there were such a solution that had
small enough degree and coefficients then it would have turned up in
searches for integral solutions such as the searches described in that ANTS-4 paper
and also on this page.
Best Answer
Yes, (with the positivity assumption) this is OEIS A025395, and members of this sequence are also found in a table on MathWorld with links to similar sequences, such as those positive integers which may be written as the sum of two cubes in exactly two ways which $1729=1^3+12^3=9^3+10^3$, famously, is an example of.