Let $y_1<z_1<y_2<z_2<\dots<y_l<z_l<y_{l+1}$ be roots of those two polynomials ($y$'s of $f=P_{l+1}^{m}$, $z$'s of $g=P_{l}^m$). I hope that they are real and alternate as written (we should think why or search it in the literature.)

Then for Wronskian we have $f'g-g'f=0$ iff $f'/f-g'/g=0$, i.e. 
$$
\sum \frac1{x-y_i}=\sum \frac1{x-z_i}.
$$
But it is impossible for real $x$. If, say, $y_k<x<z_k$ for some $k$, then $1/(x-y_i)>1/(x-z_{i-1})$ for $i=2,\dots,{l+1}$ and $1/(x-y_1)>0$, thus LHS exceed RHS.