I suppose my question could be restated as "why?" to the above statement. I'll rephrase my question if this breaks any rules asking it this way, but the expression is simply:
$$\sum_{m=2}^\infty \frac{1}{m^{2n}(m^2-1)} = ??$$
with the partial sum being: $$\sum_{m=2}^a \frac{1}{m^{2n}(m^2-1)} = ??$$
$m, n \in \mathbb N$
Are the above two somehow more insidious than they look?
On
$$ \begin{align} \sum_{m=2}^\infty\frac1{m^{2n}(m^2-1)} &=\sum_{m=2}^\infty\sum_{k=n+1}^\infty m^{2k}\\ &=\sum_{k=n+1}^\infty\sum_{m=2}^\infty m^{2k}\\ &=\sum_{k=n+1}^\infty(\zeta(2k)-1)\\ &=\bbox[5px,border:2px solid #C0A000]{\frac34-\sum_{k=1}^n(\zeta(2k)-1)}\tag1 \end{align} $$ since $$ \begin{align} \sum_{k=1}^\infty(\zeta(2k)-1) &=\sum_{k=1}^\infty\sum_{n=2}^\infty\frac1{n^{2k}}\\ &=\sum_{n=2}^\infty\sum_{k=1}^\infty\frac1{n^{2k}}\\ &=\sum_{n=2}^\infty\frac1{n^2-1}\\ &=\frac12\sum_{n=2}^\infty\left(\frac1{n-1}-\frac1{n+1}\right)\\[3pt] &=\frac34\tag2 \end{align} $$ In $(1)$, $\zeta(2k)$ is a rational multiple of $\pi^{2k}$, as shown in this answer.
Here are the first several sums: $$ \begin{array}{c|l} n&\frac34-\sum\limits_{k=1}^n(\zeta(2k)-1)\\\hline 1&\frac74-\frac{\pi^2}6\\ 2&\frac{11}4-\frac{\pi^2}6-\frac{\pi^4}{90}\\ 3&\frac{15}4-\frac{\pi^2}6-\frac{\pi^4}{90}-\frac{\pi^6}{945}\\ 4&\frac{19}4-\frac{\pi^2}6-\frac{\pi^4}{90}-\frac{\pi^6}{945}-\frac{\pi^8}{9450}\\ 5&\frac{23}4-\frac{\pi^2}6-\frac{\pi^4}{90}-\frac{\pi^6}{945}-\frac{\pi^8}{9450}-\frac{\pi^{10}}{93555}\\ 6&\frac{27}4-\frac{\pi^2}6-\frac{\pi^4}{90}-\frac{\pi^6}{945}-\frac{\pi^8}{9450}-\frac{\pi^{10}}{93555}-\frac{691\pi^{12}}{638512875} \end{array}\tag3 $$
Yes, your series is innocuous if $n$ is a non-negative integer: by partial fraction decomposition $$\frac{1}{m^{2n}(m^2-1)}=-\sum_{k=1}^n\frac{1}{m^{2k}}+\frac{1/2}{m-1}-\frac{1/2}{m+1}.$$ Hence $$\sum_{m=2}^{\infty} \frac{1}{m^{2n}(m^2-1)} = -\sum_{k=1}^n\zeta(2k)+n+\frac{3}{4}$$ where $\zeta(2k)=\frac{|B_{2k}|(2\pi)^{2k}}{2(2k)!}$ (see wiki).
However, I guess that wolframalpha considers $n$ as a complex argument (not as a non negative integer). On the other hand, given an explicit non negative integer $n$, wolframalpha will give you the result.