I’ve got a video and a photo shot in the same day and same place. But my camera (GoPro Hero 3, 720p) doesn’t allow to customise settings and video is very poor quality. I tried to fix this using curves but couldn’t achieve a good result. What approach should I use to make snow in the first picture to look like a snow in second one? I think if I will understand how to do this with one frame then I will be able to do it with a whole video. But take into account that it is considerably harder to make changes to a part of the video so it must be done with filters applied to a whole picture.
Just perform regular post-processing.
For reference, here is your first picture with all the boarder fluff stripped off:
Simply making the darkest spot black and the lightest white fixes a lot:
The snow looks a little yellowish, so here I'm using one area of the snow that looks like it should be white as the white balance reference. I also brought the dark area up a bit, although that looses a bit of the snow texture:
It's really up to you how dark you want the snowboarder to appear releative to the white snow, and how much texture you want in the snow. I'd probably continue from here by playing with the curve at the high end to get some more snow texture. But anyway, I think this answers your question well enough. Note that what I've done are all very basic and normal post processing manipulations.
For next time, you might want to take a closer look at your manual to figure out how to improve capture. Snow throws off most auto camera settings, and many cameras now have an automatic snow/beach mode to adjust (or at least an exposure adjustment setting), but if you understand what's going on even the basic adjustments available on your camera can be tweaked in your favour.
It looks like the relevant settings are in your "Capture Settings" menu (p.32 in the manual).
You probably want to turn on "Spot metering" (p33 in the manual). That will set the exposure (brightness) based on the specific object you're focusing on, not based on the overall brightness. So if you're focusing on someone in a dark snowsuit against a field of white, the bright background will have less of an impact on the exposure. How big a difference this makes will depend on how precise their "spot" is, and how much of that is taken up by dark objects versus snow and sky.
You might also want to turn on the "Protunes" mode (p36 in the manual) to access the white balance setting (p38). Set the white balance to Daylight/5500K for a sunny day, or cool white/6500K for a cloudy day.