One characteristic I found is that Fuji Pro 400H heavily favors greens and blues. The staff working at the local photo shop also confirmed this from their experience. If you take pictures in less than ideal "daylight" you can expect your pictures to shift toward green and blue. The picture below was taken just before sunset and you can see a strong green/blue shift along with some remnants of the golden sunset on the saddle bag. The corrected picture shows what it looked closer to in real life. Notice also how the shadows in the bushes behind her are just completely black. Underexposure is not your friend with this film. If I was to shoot this picture again I would overexpose at least one stop.
The hula hoop picture also shows what happens near sunset. At the time of day which this photo was taken the sun's color temperature was about 3200K, much more orange than daylight. However Pro 400H held onto skin tones and repressed warm oranges and reds. So sometimes the color balance can work in your favor too.
As mentioned before, film grain starts to become noticeable with this speed of film. The two 100% crops below show what happens in underexposed and normally exposed pictures. In my experience this is about equivalent to my old 5D mark II at ISO 3200 and with noise reduction turned on. It's not bad, but not great either.
If I had sweeping grassy landscapes, blue skies, and need to shoot until sunset, this might be a good film stock to consider. However I typically photograph people and I don't like having to correct the green out of most pictures, as well as adding saturation to skin tones. So I kept searching...