It sounds like possibly a clogged jet. If the main jet gets clogged it will start up and sometimes you can even go at a few MPH if you are careful with the throttle, but give it too much and it cuts out. Pull the carb off and look through the jets and clean everything. Might be a good time to replace the fuel filter if it's been awhile.
Probably something foreign. Small bits of a hose going bad or some junk that broke loose from the gas tank... it's hard to say. You're quite close to the symptoms of a clogged jet though, so whatever may or may not be there, I'd definitely look there first.
Find the fuel line coming from the bottom of the gas tank, you will most likely have two lines, one will be the vacuum hose and it will run into the intake manifold (area where the carb hooks to the motor). The other one will be your fuel line and it will run into the side of the carb, follow that line and there should be if one is installed a small round or cone shaped piece that is the fuel filter. If the line is one complete piece without anything resembling what I described then you will need to install one.