1) A heat soaked coil is a possibility. They absorb heat after the engine shuts off due to the stoppage of cooling air from the flywheel. You can pull the plug and with the wire on the plug, hold the hex head against something non-painted and metal on the mower. Have someone else pull the rope (it won't take much effort with the plug out) and check for a spark.
2) You could have a loose carburetor seating. Depending on the B&S motor you have, it could be a bowl type affixed to the side of the motor, or it could be screwed to the top of a metal fuel tank. Check screw/bolt tightness.
3) Carburetor could have a dirty jet. This is not very likely.
You could shoot a short burst of ether into the carb to see if it will fire off for a few seconds. If it starts and runs for a few seconds, then you know you have a fuel/airflow problem.
How fresh is your gas? Over the course of a couple years, it starts to break down and become much less useful in a gasoline engine. It may also absorb water with the ethanol content in modern fuels.