Pro #1: High thrust output. Airplanes burn A LOT less fuel at higher altitudes. The problem is that all engines produce less power as altitude is increased - most piston engines just don't have the power to lift a large aircraft to 31,000 feet or higher. A jet does. (Other advantages of high thrust are that the airplane can be larger, carry more people, go faster, and earn more money).
Pro #2: Lower weight compared to a similarly powerful piston engine.
Pro #3: Good reliability. A piston engine is much more complex. Reciprocating motion must be translated into rotational motion via a crankshaft. There are valves, camshafts, superchargers, etc that increase complexity and reduce reliability. Additionally, the propeller system of a high-performance airplane is quite complicated. A jet engine simply spins. (Contrary to what someone else said, one little pebble won't destroy a jet. They can swallow rocks, ice, birds, and/or turbine blades and still keep spinning - you'll have a large repair bill but the engine will most likely continue to operate).
Pro #4: The fuel for a jet engine is less refined and less expensive than aviation gasoline.
Con #1: Longer "spool up" time. Although they're getting better the power of a jet engine is not immediately available. (You move the thrust lever and it takes several seconds for the engine to accelerate). This isn't usually a problem unless you encounter windshear or have to execute a go-around. Power in a piston comes much more quickly.
Con #2: The prop provides a lot of airflow over the wing (assuming a multi-engined aircraft). If you get too slow in a prop as soon as power is applied the prop provides an immediate increase in airflow over the wing which is extra insurance against stalling. In a jet the entire aircraft must be accelerated before any benefit is realized. This takes much longer.