I have an LSH-S, have used it extensively, and recently added the LSHF-P. No LSH-P, so my remarks may not be of full value.
The best choice really depends on what use the light gets.
#1 use: Night running companion/guide. I run 3 or 4 times per week, usually late at night on account of schedule limitations. The LSs help me through unlit sections and spotlight aggressive dogs until I'm past. The LSH-S gives me a more "daylit" look, but I choose the LSHF-P since it stops the dogs better.
#2 use: At work to inspect construction, investigate field conditions, and work in unlighted building areas. The LSH-S is better for picking through an unlit building, but the LSHF-P is my choice because its excellent throw saves me ladder trips when investigating conditions above ceilings.
#3 use: General purpose after dark light. The LSHF-P casts a tighter overall beam than the LHS-S, yes, but it's surprising how often the beam it casts is exactly the right size, and the sharp cutoff is sometimes actually a benefit. Just last night I was helping my niece hunt for something in her car's trunk. No trunk light. From my distance the beam nicely matched the trunk mouth size, there was plenty of light for her to poke around, and the sharp cutoff kept the light from glaring in her eyes.
#4 use: Camping. Dead heat. The LSH-S wins for general camp strolling since it gives a more natural look to the woods ahead. The LSHF-P wins when spotting wildlife and distant rambunctious children.
I was at first sceptical of the LSHF-P's versatility, but it has really grown on me. The beam width is an excellent balance of spread and throw. Wide enough for general purpose use, even without strong spill light. Narrow enough to throw further than the NX05.
But...all of this is splitting hairs. Either of the lights work very well for any of my uses. I should mention that I do not use either LS for long duration close tasks, such as night reading. I use an Arc AAA or AA for that.