Take a seat...
It's often easy to spot the differences between a gaming chair and an office chair. For one, the bright colours and often larger frames of gaming chairs mean they stick out immediately whereas office chairs can blend into the background far easier.
A gaming chair is usually designed in the racer style, with a tall winged back, bucket seat and raised seat edges. You'll often usually find lumbar and head support cushions included, with armrests that can be adjusted in a minimum of three directions. You'll also find that gaming chairs can recline significantly further than office chairs, and offer rocking and tilt locks as well.
An office chair tones things down considerably. You'll find a flat cushion with a wingless back, often with fixed lumbar support built in. Armrests will generally only move up and down, and reclines will be limited with more restrictive tilt locks.
Gaming chairs are built to withstand longer use, which means they often employ materials that more users find comfortable. You're more likely to find memory foam seat cushions in the starting price category, with padding all the way up the back and on the armrests as well.