Vadim, that's a good question. The difference between "smoking" and "lighting up" is that "smoking" refers to someone doing the action. That is, the cigarette is already burning and they are smoking it.
"He is smoking in the car"
"She smoked as she talked on the phone"
"Lighting up" refers to the action of starting a cigarette to burn. For example,
"He took a cigarette from the box and is lighting it up now"
"He lit up a cigarette as he left the building"
You may also find a related phrase "light it up" used. For example in some war movies or books when they begin to shoot or bomb something.
Simply, "smoking" is used when something is burning. "Lighting up" is used to describe the action of starting something to burn.
I hope this helps.