For a one-sided, unbounded distribution when you still want to observe changes without being susceptible to outliers.

If you're monitoring response timings on a server, for example, the median might be very close to 0, and it won't shift unless a majority of the distribution slows down. If you take a winsorised mean, you can trim useless long response times that mess with the mean, but still see if e.g. 1/3 of your responses are suddenly slower than normal.

When processing astrophotos, multiple exposures are “stacked” together. There is a certain amount of noise in each frame - due to electronic noise or simply the random number of photons that strike a pixel in any one exposure - that you want to average out on a per-pixel basis.

However some frames may contain unwanted outliers, for example if a satellite briefly passes overhead it will appear as a very bright streak in only one frame.

By winsorizing, outlying pixel values can be eliminated while still maintaining the same number of samples per pixel as the rest of the stacked image.

A lot of financial data is of dubious quality. Lot's of spurious data points, mostly in the extremes, just aren't real. Median is useful for many things, but sometimes you just have to get rid of the bad data to get a valid mean.

Any of: low amount of data points; wanting a continuous value if your points are integers; slightly different behavior in high dimensional vector spaces.