This direct screenshot from Wikipedia shows the conversion between Lab and HCL.
HCL is only for convenience, it doesn't change the color space (except in case the numbers are limited to sparse integer values)
For ex. I like to use HCL version because I can imagine the color changes more easily beforehand by adjusting hue and chroma of HCL. Trying to guess how to change a and b in Photoshop's Lab system for a wanted change is much more trial and error than using HCL which is available in GIMP.
Fortunately in Photoshop I can also adjust hue and chroma. The hue/saturation adjustment works with hue and chroma in Lab mode, but the colors are in the color picker shown as Lab-numbers.
I prefer HCL over Lab if I can choose because the bones of my head are too thick for intuitive handling of Lab-numbers. But there exists people who have things differently. See for ex this old case:
Is there a way to remove lens flare?
Hopefully you have already noticed that Lab and HCL are made to present the colors which humans can see so that the luminance, chroma and hue are visually separated from each other. Actually in Lab hue and chroma are mangled together in a and b, but L is independent.
RGB and its polar variations (which are made also for covenience) present commands for colored light production machinery in computer screens and what RGB sensors in a camera output. The visual effect of the saturation in polar RGB systems like Photoshop's HSB depends drastically on hue and brightness. In HCL the factors are much more visually independent. That has an unfortunate consequence: It's far too easy to select in Lab or HCL a color which cannot be produced in RGB screens. Fortunately the programs have options to switch out of gamut warning ON.