A wok and a skillet cook almost identically on a home stove, because the thing that actually separates them — a wok's sloped wall staying hot enough to sear — depends on a burner most kitchens don't have. Buy a wok for its shape and its other advantages. Don't buy it expecting restaurant heat.
Why a wok's walls need heat the stove can't give them
A gas range's largest burner puts out somewhere around 12,000 to 18,000 BTU an hour, and the ring of flame it produces is sized to heat a flat pan bottom sitting close above it. A commercial wok burner is a different piece of equipment entirely — commonly rated at 100,000 BTU or more, with the flame shaped to wrap up around a wok's curved base rather than sit flat underneath it. That's what heats the wall, not just the bottom.
Set a wok over a home burner and the flame only reaches the small, flattish patch at the center of the base. The curve above it sits mostly outside the flame's ring, so it never climbs anywhere near the heat a stir-fry needs to sear food instead of steaming it.
What that does to the technique wok cooking is built around
The point of tossing food up a wok's sloped side is to park it somewhere hot but less intense than the bottom — sear at the base, hold at the wall, cycle ingredients between the two so nothing overcooks while everything gets a turn at high heat. Take the hot wall out of that equation and tossing food up the side of a home wok does nothing except move it away from the only heat source in the pan. It cools. A skillet does the same thing when you push food to one side; the wok's slope just makes the motion easier.
Where the shape still earns its place at home
Two real differences survive the move to a home stove, and neither is about temperature. A wok's sloped wall gives more usable volume for the same footprint on the burner, so a big batch of fried rice or noodles has somewhere to go when you toss it instead of sailing over a skillet's straight side. And most woks are thinner steel or carbon steel than a heavy skillet, so they heat up and cool down faster — useful for the quick highs and lows a stir-fry recipe calls for, even though the top of the pan never gets as hot as the bottom does.
Flat bottom or round bottom, on a home burner
A flat-bottomed wok sits directly on a gas grate or electric element and puts more of its base in contact with the flame ring than a round-bottomed one would, which touches at a single point and needs a stand just to stay upright. Neither shape solves the wall problem — contact area at the bottom isn't the same as heat climbing the side — but a flat bottom is the more practical choice on a stove that wasn't built for the other kind.