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I was going to ask "Are there any missions where a geosynchronous orbit is advantageous over a geostationary orbit?" The TL;DR answer to this question (Are there any satellites in geosynchronous but not geostationary orbits?) is "Yes", however, the answer there doesn't explain what any of the benefits are.
What are the benefits? Why would a mission choose a geosynchronous orbit over a geostationary orbit?
Here are some examples:
when the ground terminals on Earth use a low enough frequency that they are more or less omni-directional and don't need to point exactly at the satellite. In such a case it doesn't matter so much if the satellite is not at 0.0 inclination. This has typically been done with L-band/S-band. The benefit is saving the cost of the station-keeping thrusters/propellant.
when the ground terminals on Earth are large enough that they can auto-track the satellite. This clearly brings up a trade-off of the cost of doing so over all ground terminals vs the cost of getting the satellite to station-keep at 0.0 inclination. The autotrack solution is rare (maybe not chosen at all?) these days, it was more common in the dim and distant past.
when several satellites are kept in the same slot at high inclination in order to achieve high latitude coverage. The idea is that there should always be at least one satellite in view. This is the scheme used by the QZSS (Quasi-Zenith Satellite System) in conjunction with example 1 above.
