This seems to have gotten off course; I have used lawn tractors and garden tractors and test drove a zero turn (poorly); my opinions follow.
The lawn tractor is a good riding lawn mower with limited other uses such as pulling a cart, etc. Depending on the make, it may or may not have the ability to mount ground engaging tools. Usually the lowest cost.
The Garden tractor is heavier duty, with a higher price tag. Besides mowing and towing, there is usually a rototiller and or moldboard plow offered, front blades, a snowblower, etc. Some even offer a front bucket. Kind of the McGuyver of small tractors.
The zero turn is a specialized tool for grass cutting. Manueverability and speed of cutting with a price tag. Limited utility.
Obviously I favor the garden tractor. Slightly heavier contruction, more power, more versatility.
As an example, right now, if I got my place in the country, I'd like a JD 310 (hydrostatic, power lift and power steering) with a mower, cart, and the front blade for light landscaping and the occasional snow we get in SE Virginia.