There's a related question, but it doesn't seem to apply to spacecraft trajectories.
Suppose you want to send a non-thrusting spacecraft far from earth, and you have ground tracking of the spacecraft, but the spacecraft doesn't have any way of directly incorporating position/velocity updates into its nav system (no GPS, for example). So, you're getting range and range-rate updates on the ground and incorporating them into a trajectory approximation using a batch least squares estimator; and then you're propagating it forward in time using your knowledge of the system dynamics. You want to transmit this estimated trajectory to the spacecraft so that it can interpolate its position and velocity instead of relying on a noisy IMU.
I've heard of people using Chebyshev polynomials for this purpose, but it seems like SPICE has much better support for Hermite polynomials, and makes it much easier to produce Hermite-based spacecraft ephemeris than Chebyshev. While JPL has published a method for producing Chebyshev position/velocity fits for planetary bodies, the paper does not discuss their applicability to spacecraft or satellites.
In general, are there reasons to choose Hermite over Chebyshev, or vice-versa? I can't really find any discussions of their pros and cons, even in the SPK required reading from NAIF/SPICE.