For the record, disregard Kirk, he has not a clue. Jets do not turn 100,000 RPM, and commercial airliners glide so well often the problem is to get them to descend quickly when needed.
Unless you are talking about a 70 year old DC3 flying cargo into the bush, all "prop" airliners you will see are turboprops, basically the same engine that powers a jet, but turning a prop instead.
A turbofan can fly higher (and faster) than a turboprop, so there is the ability to avoid a little more of the weather, but turboprops normally require less runway to takeoff and land and burn less fuel at lower altitudes. As far as basic safety goes, there's not a huge advantage ether way.
The real question is not jet vs prop, it's piston vs turbine. A turbine engine, whether in a turbofan (a jet) or a turboprop, is simpler and more reliable than a piston engine. In small personal planes, twin engine piston planes are almost extinct, being replaced in the market by single engine turboprops. Small twin engine planes, like the Cessna 310, have a worse safety record than single engine planes after an engine failure.
A single engine plane, whether piston or turbine, is required to have a pretty low glide and landing speed, so if the engine quits you have a good chance of making a safe landing. It turns out having one turbine engine that might fail once every 2,000,000 hours is safer than having 2 that might fail once in every 100,000 hours.
As to your edit, there is no such thing as a "propeller engine" - you can use any type of engine or motor that spins to turn a prop. A turbine spinning a prop is much the same as one spinning a fan in a jet, or a rotor in a helicopter. Did you know that the core of the jet engines they use in "jet dragsters" usually do not come out of jet planes, but out of helicopters?
A turbine engine is basically a set of spinning disks, mechanically much simpler that a piston engine with parts moving back and forth, a motion that must be converted to a rotary motion, and with valves that must open and close thousands of time a second.