  We study Diophantine equations of type f(x)=g(y), where both f and g have at
least two distinct critical points and equal critical values at at most two
distinct critical points. Some classical families of polynomials (f_n)_n are
such that f_n satisfies these assumptions for all n. Our results cover and
generalize several results in the literature on the finiteness of integral
solutions to such equations. In doing so, we analyse the properties of the
monodromy groups of such polynomials. We show that if f has coefficients in a
field K, at least two distinct critical points and all distinct critical
values, and char(K) is not a divisor of the degree of f, then the monodromy
group of f is a doubly transitive permutation group. This is the same as saying
that (f(x)-f(y))/(x-y) is irreducible over K. In particular, f cannot be
represented as a composition of lower degree polynomials. We further show that
if f has at least two distinct critical points and equal critical values at at
most two of them, and if f(x)=g(h(x)), where g and h have coefficients in K and
g is of degree at least 2, then either the degree of h is less or equal than 2,
or f is of special type. In the latter case, in particular, f has no three
simple critical points, nor five distinct critical points.
