It is not an answer, rather long comment...
I am sorry: my previous posts were incorrect, I will correct below.
I would suggest you guys insert the statement and may be proof to Wiki article, it is quite worth and since it was mainly written by me, imho I might give such a suggestion.
The main message is that there is "certain relation" (described below) between standard Gelfand-Tsetlin maximal commutative subalgebra in $U(gl_N)$ and the maximal commutative subalgebra in $C[S_N]$ generated by Jucys-Murphy elements. The relation consists of two steps which can be seen as generalized Schur-Weyl duality and generalized $gl_M - gl_N$ duality. Both steps involves an intermediate object - "bending flow" commutative in $U(gl_N \oplus ... \oplus gl_N)$ (sum contains $M$ terms). Briefly speaking these generalized dualities say that: images in certain representations of these commutative subalgebras coincide.
Since I forget some details I would NOT make again the claim that "JM elements go to "quadratic Casimirs"", which might give another (but very long) way to answer Igor's question. Just simply describe the relation which might be interesting on its own.
Step 1. Generalized Schur-Weyl from JM to "bending flows". (Rather trivial step).
Consider $V=C^N \otimes ... \otimes C^N$ ($M$ terms in tensor product). $C[S_M]$ acts here in a natural way. $U(gl_N \oplus ... \oplus gl_N)$ surjects on $End(V)$. Since it surjects we can find certain elements in $U(gl_N \oplus ... \oplus gl_N)$ which are mapped to JM elements, moreover we require such elements to be quadratic in generators of $U(gl_N \oplus ... \oplus gl_N)$, and it would fix these elements. The basic idea is that the permutation operator (12) acting in $C^N\otimes C^N$ is OBVIOUSLY an image of $\sum_{ij} E_{ij}\otimes E_{ji} \in U(gl_N)\otimes U(gl_N)=U(gl_N\oplus gl_N)$ and nothing more than that.
By $E_{ij}$ denoted the matrix with $1$ at position $(ij)$ and zeros everywhere else.
So we get certain commutative subalgebra in $U(gl_N \oplus ... \oplus gl_N)$ such that it is "Schur-Weyl dual" to JM subalgebra, meaning that the images of these subalgebras in $End(V)$ coincide.
Step 2. $GL_M-GL_N$-duality from "bending flows" to Gelfand-Tsetlin. (This step is not so trivial). It is mainly due to Flaschka and Millson - section 8 of http://arxiv.org/abs/math.SG/0108191
Consider the vector space $W = S(C^N\otimes C^M) = S(C^N \oplus ... \oplus C^N)$ (M-terms in sum) and $S$ denotes symmetric algebra of the vector space. Lie algebras $gl_M$ and $U(gl_N \oplus ... \oplus gl_N)$ acts on $W$ in a natural way.
Theorem: the images in $End(V)$ of GT and "bending flows" coincide.
In such a form it is Theorem 2 page 9 in our paper: http://arxiv.org/abs/0710.4971
Why the name "bending flows" ? If we make similar considerations for $U(so_3 \oplus ... \oplus so_3)$ or more precisely its associated grade Poisson algebra $S(so_3 \oplus ... \oplus so_3)$ we get a (Poisson) commutative subalgebra there.
The beautiful fact is that "JM" type generators have very nice geometric interpretation. We can identify $so_3=R^3$ and so elements of $(so_3 \oplus ... \oplus so_3)$ can be seen as $M$-gons in $R^3$. The statement is that if we "bend" polygon along the non-intersecting diagonals then such flows will be hamiltonian and will be defined by JM-type generators in $S(so_3 \oplus ... \oplus so_3)$. Well, I omitted some details and may be comment is not so clear, one should draw simple pictures in order to see what is going on.
Bending flows were proposed for $S(so_3 \oplus ... \oplus so_3)$ in paper M. Kapovich, J. Millson, The symplectic geometry of polygons in Euclidean space,J. Differ. Geom. 44, 479–513 (1996)
Generalized further in several papers in particular in Gregorio Falqui, Fabio Musso, Gaudin Models and Bending Flows: a Geometrical Point of View, J. Phys. A 36 (2003), no. 46,11655–11676. nlin.SI/0306005