In my region is it usual that we say 'white' for hair which is not black or brown anymore as the age passes. But I've recently come to know that the word 'grey' can also be used instead of 'white'. Which word do you native use? The colour that I mean to ask about can be seen in the image.
In America, at least, "gray hair" is a catch-all category that includes white. It would not be surprising to refer to someone with white hair and beard as having gray hair.
Still, the more precise description of the beard in your picture would be white. Santa Claus is always depicted with white hair and a white beard, and almost nobody thinks of that image as being in any way gray. (And remember, the main difference between Gandalf the Grey and Gandalf the White in Peter Jackson's film series of The Lord of the Rings was the color of his robes.)
It really just depends on how fussy you are. Hair color, like eye color, can be hard to pin down. A redhead I dated once asked me to describe her hair color. I told her it looked orange to me. She was pleased, and told me I was the first one who didn't just automatically tell her it was red. (For the record, it really was orange.)
N.B. "Gray" and "grey" are alternate spellings of the word, the latter being chiefly British. But it doesn't matter which you use, unless you're trying to spell a proper name.
This may be my idiosyncratic take on it but I think grey hair and white hair are different, at least when talking about many hairs. For individual hairs, I would use the terms interchangeably.
When all the hair on someone's head is grey / white, I would describe them as having white hair. But as long as they still have an appreciable number of darker hairs, I would say their hair is grey.
It's the obvious thing. Grey and white are just colours and you use whichever colour is most appropriate: "grey" if it's grey and "white" if it's white or very pale grey. How pale is "very pale" is a subjective decision and will probably depend on the light, anyway. If you do image searches for "grey hair" and "white hair", you'll see that most of the images for "white hair" are significantly paler than most of the "grey hair" images.
There are a lot of very good answers here. The guy in the picture appears to me to have white rather than gray hair. Though as @Flydog57 says, the verb that describes the process of hair lightening with age is graying (or greying).
Also, if you want to be literary, you can use the word “hoary” to describe either the old person or their hair (not recommended for everyday speech, though):