Here's my take on the matter: the difference of philosophy between quadrature routines and ODE solving routines, I believe, is this:

Extrapolation is riskier than interpolation.

Remember that numerical quadrature routines all boil down to approximating your presumably more complicated integrand within the interval of integration with something easier to integrate exactly, and then integrating that. For instance, Newton-Cotes (and in essence, Romberg as well) constructs an interpolating polynomial from your integrand with equispaced abscissas, and integrating that. For Gaussian or Clenshaw-Curtis quadrature, it is equivalent to interpolating your function at "specially spaced" abscissas (Legendre polynomial roots in the former, and Chebyshev polynomial roots in the latter) that have better convergence in the limit. In effect, we run under the assumption that the interpolating function behaves very similarly to the actual integrand within the interval of interest that a sufficient amount of sampling within the integration interval should be enough to capture the behavior of your integrand, and thus give a result hopefully close to the actual value of the integral.

In contrast, remember that ODE solvers usually only have initial values to start with. The reason for building in a lot of machinery in current ODE solvers, whether Runge-Kutta, Bulirsch-Stoer, Adams/Gear multistep, or some of the fancier modern techniques, is that *extrapolation is inherently unstable*. Knowing how the solution looks like in the beginning gives no guarantee how it will behave as the ODE solver marches on; the solution may well be violently oscillatory, or decaying quite fast (so-called "stiff" problems). Thus, there is quite a fair amount of code inscribed in modern ODE solvers for checking how reasonable are the step-sizes being taken, and other such fail-safes.

As I did mention in some previous comments, some ODE solving methods are equivalent to quadrature methods when applied to the initial-value problem $y^{\prime}=f(x)$: using classical Runge-Kutta for quadrature is equivalent to performing Simpson's rule, for instance.

The point is that ODE solvers tend to be more careful ("tiptoeing", if you will) and thus more effort-intensive than numerical quadrature routines because they make no assumptions on how your integrand behaves. On that note, I will say that you should know that there are integrands (and corresponding intervals) where using an ODE solver might make more sense than using a numerical quadrature routine. One instance that comes to mind: if you know (through graphing, for instance) that your integrand has crazy behavior *in a relatively tiny interval within the interval of integration*, while the sampling done by a numerical quadrature routine might miss such features (or take a long time to notice them), an ODE solver will be careful to begin with and is less likely to miss the crazy behavior, and will shrink step-sizes as appropriate until it has gone past that interval.