Robert describes the differential equations which one can write in terms of $\tau$ and $\kappa$, and the inherent limitations in this *local* approach. But maybe one can find more reasonable or useful conditions in terms of *integral* equations, and the whole problem could be much more interesting or natural if we consider *closed* curves. In other words, a *global* approach could be more enlightening.

For instance, a necessary condition for a closed curve to lie on a sphere is that $\int \tau=0$, e.g. see p.171 of [Millman and Parker][1], which incidentally turns out to characterize spheres. Furthermore, any closed curve lying on a convex surface must have at least $4$ points where $\tau=0$, which is a generalization of the classical four vertex theorem due to [Sedykh][2]; see also this [paper][3] for another proof, and this [paper][4] for a generalization. Another necessary condition for a curve to lie on an ellipsoid is that it have a pair of parallel tangent lines, which turns out to characterize ellipsoids, as described in this [paper with Bruce Solomon][5]. 

It would be interesting to find more simple or nice necessary conditions for a closed curve to lie on an ellipsoid, and I think it is possible that a collection of these may turn out to be sufficient as well.

**Addendum 1:** I found a nice paper which seems to be relevant:

Kreyszig, Erwin; Pendl, Alois
[Spherical curves and their analogues in affine differential geometry.][6] 
Proc. Amer. Math. Soc. 48 (1975), 423–428. 

In this paper the authors define a curve to be spherical in the affine sense if all its normal planes pass through a common point. If I am not mistaken these include curves which lie on ellipsoids, but other curves as well. At any rate, they obtain a very nice characterization for these "affine spherical curves":
$$
\left(\frac{1}{\widetilde\tau}\right)''+\frac{\widetilde\kappa}{\widetilde\tau}=0,
$$
where $\widetilde\kappa$, $\widetilde\tau$ are the affine curvature and torsion and diferrentiation is with respect to affine arc length. 

**Addendum 2:** Maybe a necessary condition could be that $\int\widetilde\tau=0$, but this is just a quick guess, as I am no expert in affine differential geometry.

  [1]: https://www.amazon.com/Elements-Differential-Geometry-Richard-Millman/dp/0132641437
  [2]: https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms/26.2.177
  [3]: https://arxiv.org/abs/1704.00081
  [4]: https://arxiv.org/abs/1501.07626
  [5]: https://arxiv.org/abs/math/0205222
  [6]: http://www.ams.org/journals/proc/1975-048-02/S0002-9939-1975-0365369-0/S0002-9939-1975-0365369-0.pdf