The short answer is that both will work, however, if you are looking for high accuracy then a screw drive system will outperform every time.
Belt-Drive Systems
Belt drive systems are much cheaper to produce and require less expensive components to make them work. A belt drive system is usually a one to one ratio between motor movement and gantry movement. Meaning that there is no gear reduction so as the motor moves, the gantry will move an equal amount. Because of this, the motor on a belt drive system will usually be much larger than a same size lead screw driven system. Belts are also more susceptible to stretching and wear. This will always affect accuracy. If the belt stretches that will immediately translate into an inaccuracy at the bit into your workpiece. Some might argue that a belt drive system will move the gantry faster, but keep in mind that moving faster into a material and having stretch on the belt will cause deflection and the belt will act more like a spring thus increasing the risk for lost steps and positioning.
Screw-Drive Systems
A screw driven system usually consists of a threaded rod (screw) that is either an Acme-type lead screw or a ball screw. The motor will rotate the screw and the gantry is attached to the screw with some type of threaded nut (lead screw nut or ball nut). As the screw turns, the gantry is moved in the corresponding direction. The number of threads per inch will determine how many rotations the screw must turn for the gantry to move one inch. This gear reduction means that you can either use much smaller stepper motors to control the movement and still maintain a very high accuracy, or you can use a larger motor and move a larger mass as compared to using the same size motor in a belt driven system. You will find much better repeatability and accuracy in a screw driven system with minimal to no linear free play. Due to the costs of lead and ball screws and components, you will find that these type of drive systems will cost more than a belt drive system.
Summary
If you are looking for high accuracy, a more rugged design, or if you plan to cut harder materials like hard plastics, hard woods and metals, then you will want a system that uses a screw-drive. If you are just planning to cut thin, softer materials like soft woods and plastics and accuracy is not as important to you, and if you do not have a large budget, then a belt-drive system might be right for you. Always keep in mind when looking at purchasing a CNC system what you might need in the future. You might think that your immediate need to make model parts out of thin plywood, thin plastic and balsa wood might not require a screw drive, but maybe in the future you want to cut hardwoods, thicker plastics or metals, or you get a project that requires a high cutting accuracy – in this case you might be kicking yourself for not spending the extra few hundred dollars for the better drive system.