Diagnosis
With good spark and gas, the engine should at least cough. The fact it doesn't points to a lack of fuel actually making it into the cylinder. On these, the most likely cause is a clogged pilot jet or passage in the carburetor from old fuel residue.
How to Fix It
You'll need to clean the carburetor. Remove the air filter cover and the filter. Then, disconnect the fuel line (pinch the spring clamp). Remove the two bolts holding the carburetor to the engine. There's often a gasket between them. Once off, remove the float bowl (single bolt on the bottom). You'll see a small, main jet and an even smaller pilot jet (sometimes called the "idle jet"). Use a can of carburetor cleaner and a fine strand from a wire brush or a dedicated jet cleaning tool to clear the tiny hole in the pilot jet. Spray cleaner through all the small passages in the carb body. Reassemble everything.
If That Didn't Work
If it still won't start, check that the choke plate is actually closing fully when you move the lever. If the linkage is bent or disconnected, the engine isn't getting the rich fuel mixture it needs to start cold.
Heads Up
When reinstalling the carburetor, make sure the small spring that connects the governor linkage is hooked back up correctly. If it's not, the engine will either not run or will immediately rev to dangerous RPMs when it starts.