I have a 3.75 hp Briggs & Stratton Sprint engine on my push mower that has just recently begun to sputter and not get up to speed. I've cleaned the carburetor and fuel tank, replaced the spark plug and replaced the sponge air filter, but it still sputters. Oil is kept at proper level.

I shot a video of it sputtering with the air filter removed so there is a clear view of the carburetor. I have little to no small engine repair experience, so please type slowly. http://youtu.be/1OOd5MfELsI

http://youtu.be/1OOd5MfELsI

Are you certain of the right gap on the spark plug?
Also, maybe research governor problems and check it in the dark for possible bad plug wire arcing to ground.
Fresh gasoline?
I had a similar problem a few days ago. Air filter was clogged, which I replaced but no improvement. Removed plug which looked good but gap had "grown" to .040 and supposed to be .030. Once reset gap, problem was solved. Good luck.

PS: just read a report of a guy who said he found a bad diaphragm between tank and carb-?

Hi,
Watched your video. Be aware that you should have the screw for the air cleaner in place when running. My assessment is that you need a diaphragm, part #495770(item #394) and you need to hook up the governor spring correctly; item #209 should connect in the loop of item #201 link. Pictures attached(if you supply the model number(6 characters) and the type/trim(also 6 characters) I will link you to the entire parts breakdown.
Peace,
Clarke