New Briggs & Stratton El600 push mower / initial checkout 

2 things you can also try on the old one: 1) Air Filter. They get plugged easily on these machines. 2) Muffler. I've had a few plugged on this style due to the filter getting plugged and running rich. A quick look at the spark plug might show some carbon build up too. Given they're the same engines, they are interchangeable so might be some easy quick testing. If the muffler is plugged, you can just heat it up with a torch to help break down the build up.

Have you tried to clean or rebuild the carb on the old mower?