On my flight bag, I have a sticker that says "Dash Trash" in large type and then "Silly Pilot, Jets are for Hot Tubs!" It's a (now ex-) turboprop driver's playful rejoinder to the jet jocks' standard jab that "Props are for boats." The war of words between prop pilots and jet drivers has been mostly waged in jest; just about everybody understands that however cool they look and sound, modern jets aren't harder to fly and don't confer superhuman status on their pilots. That said, there are some key differences between jets and propeller driven aircraft that anybody making the transition for the first time must keep in mind.
The powerplant itself is actually one of the more minor differences, at least when transitioning from a turboprop to a modern jet. Most of today's "jets" are actually turbofans and little of their thrust is true "jet thrust" caused by exhaust gasses being expelled from the tailpipe. Rather, the majority of the thrust is generated by the large fans on the front of the engine, very similarly to how propellers are used in turboprops. The GE90 engine used in the B777, for example, has a 9:1 bypass ratio, meaning that for every pound of air that moves through the engine core and is used in combustion, nine pounds of air are accelerated by the fan blades and bypass the engine core. Similarly, the PW150A turboprop on the Q400 gets 85% of its power from the propeller; the remaining 15% is supplied by jet thrust from the engine core. You could almost say that modern jet engines are really ducted turboprops. The main difference is that turboprops have fewer blades of much larger diameter that are geared down to slower rotational speeds. This makes them more efficient at lower speeds and altitudes but limits high speed potential because the propeller tips encounter supersonic speeds long before the aircraft is supersonic. In turbofans, the duct and fan blades/low pressure compressor cause an area of high pressure that slows incoming air enough that the aircraft can cruise at transonic or even supersonic speeds while providing steady, subsonic airflow to the engines.
Operationally, the main difference in powerplant operation has traditionally been spool up / spool down time. A piston engine can usually go from idle to full power in very little time. Normally aspirated piston engines just need more fuel/air mixture to accelerate, and this is available instantaneously. Jet and turboprop engines need more compression, which is obtained by adding fuel to the combustion section, causing the turbines to spin faster, which drives the compression section faster, increasing the mass of air available to the engine for increased power. There's some lag involved. On the first generation of turbojets, it could take as long as 10 seconds to go from idle to full thrust, which caused several accidents when pilots inexperienced with jet engines found themselves at low altitude and low airspeed with "unspooled" engines. Adding another spool to the engines so there were separate low pressure and high pressure compressors helped considerably but lag was still a factor so long as most of the thrust was generated by accelerating hot gasses out the tailpipe. With high bypass turbofans, however, thrust increases as soon as the low pressure spool accelerates. Spool-up time can still be somewhat lengthy - it varies by design - but the engine doesn't need to be fully spooled up to produce significant power. Turboprops still hold an edge in providing near-instantaneous power but the difference isn't as much as it used to be.
Now this is not to say that going from turboprops to jets isn't a significant transition. It is, but generally for reasons not directly related to the powerplant. Most of the differences are in aircraft systems, high speed/high altitude aerodynamics, and low speed swept-wing characteristics.
Compared to older turboprops like the King Air, Metroliner, or J31, most jets are more capable, more automated, and designed with greater redundancy. Over the long run this makes life a lot easier on the pilot, but during the transition there are significant systems differences to be learned. Some systems like the hydraulics or pneumatics may be more complex, unfamiliar technologies like fly-by-wire may be incorporated, and the avionics (and the electrical system that powers them) will be more important than anything else in the airplane. This really isn't a prop vs jet issue; it's an old plane vs new plane issue. Recent turboprop designs like the Q400 and Piaggio Avanti utilize technology to about the same degree as jets of the same vintage.
Like I said, the aerodynamic limitations inherent to propeller-driven engines mean that they typically operate at lower altitudes and slower airspeeds than jets. Therefore, the transition to jet aircraft also involves learning about high speed and high altitude aerodynamics. Most transport-category jets regularly cruise at speeds of .75 to .85 Mach (75%-85% the speed of sound). This is within the transonic speed range, meaning that while the aircraft itself is below the speed of sound, airflow over some parts of the aircraft will be accelerated to supersonic speeds. This typically isn't much of a problem until airflow over the wing reaches supersonic speeds, which is known as Critical Mach or Mcr. At Mcr, a shock wave begins to form on the wing; as speed is increased beyond Mcr, the shock wave intensifies and moves aft, which has several undesirable side effects. The shock wave moves the wing's center of pressure aft, which requires increasing amounts of elevator to counteract. Eventually, the elevator may run out of authority and the aircraft will pitch down, which increases airspeed and exacerbates the situation, a phenomenon known as mach tuck. Additionally, the shock wave will cause flow separation which decreases aileron effectiveness and, depending on aircraft configuration, may blanket the horizontal stabilizer and contribute to the tendency towards mach tuck.
The problems associated with transonic flight have been well known for 60 years and aircraft designers have a number of weapons in their arsenal to combat them. The use of swept wings is universal among aircraft designed to cruise above .70 Mach; it delays critical mach by artificially increasing the wing's fineness ratio, since the relative wind travels a greater distance across the wing than its actual chord. Supercritical wings, which have a relatively flat upper surface and an unusually curved lower surface, decrease the strength of the shock wave that forms on the upper wing at speeds beyond Mcr. Vortex generators add energy to the boundary layer and increase resistance to flow separation in both high speed and low speed flight. Trimable horizontal stabilizers provide the elevator authority necessary to counteract the aftward shift in center of pressure. All these design features can decrease or delay undesirable effects of supersonic flow but they do not eliminate them. For this reason, new designs are thoroughly flight tested and a maximum operating mach limit (Mmo) is established. This is in addition to Vmo, the maximum operating airspeed limit, which provides structural rather than aerodynamic protection. Typically Vmo is limiting below FL250-FL300 and Mmo is limiting above those altitudes.
If high-speed aerodynamics were your only worry at high altitude, staying out of harms way would be a snap: don't exceed Mmo. However, low-speed aerodynamics also come into play. It's somewhat counterintuitive, because you're zooming over the ground at 400+ knots, so you'd think you wouldn't have to worry about stall speed. Keep in mind, however, your indicated airspeed will be quite low compared to true airspeed when at high altitudes. Angle of attack varies with indicated airspeed, meaning you're much closer to a stalling AoA in high-altitude cruise. If you plot maximum and minimum airspeed versus altitude on a chart, you'll see the two lines eventually merge as altitude increases. The area where there is little margin between high speed and low speed danger zones is colloquially known as coffin corner. Keep in mind that the danger of stalling comes not from low airspeed but high angle of attack; therefore, an altitude that's perfectly safe for level flight at a light weight in smooth, cold air could be dangerous for maneuvering, at a heavier weight, in turbulence, or at higher temperature. You can avoid coffin corner by using your aircraft's maximum cruise altitude performance charts: they'll show you the highest altitude you should cruise at given a certain aircraft weight and air temperature.
One final consideration is the handling characteristics of a swept-wing jet. Sweeping a wing introduces significant spanwise flow at low speeds, making the wing tips susceptible to stalling before the roots. This condition is unacceptable for a few reasons ranging from loss of lateral control to the forward shift in center of lift which causes a further pitching up moment. Aircraft designers use a few tricks like washout, cuffs, or vortex generators to keep the tips from stalling first, but the fact remains that swept-wing aircraft seldom stall as docilely as straight-wing airplanes do. The solution is to never allow the aircraft to stall, and to this end most swept-wing planes have stall protection devices like stick shakers and stick pushers. That said, you don't need to stall a swept-wing jet to get in a pickle; getting slow at low altitude will do nicely, as the spanwise flow also guarantees rapidly increasing induced drag below Max-L/D. When jets are lost in landing accidents, it is often due to an unarrested low-speed sink that results in impact short of the runway.
Lastly, some swept wing aircraft are susceptible to "dutch roll," a phenomenon of simultaneous uncoupled roll and yaw oscillations that progressively increase in magnitude. It's most prevalent in aircraft with strong lateral stability; swept wings are inherently stable laterally. Basically a sideslip causes extra lift on the side of the slip, resulting in a rolling moment in the other direction at the time that directional stability is causing yawing moment in the original direction of the sideslip. If the directional stability is weak compared to lateral stability, the yawing moment lags slightly behind the rolling moment, allowing an opposite sideslip to develop, and the process repeats again, feeding on itself. In extreme cases it can render an aircraft uncontrollable; susceptible designs prevent it by installing a yaw damper. If your yaw damper can be deferred (MEL'd), your aircraft probably isn't extremely prone to dutch roll, but you'll want to make a mental note to stay coordinated with an inop yaw damper rather than flying with your feet on the floor like usual!
None of this is very hard stuff to grasp and the procedures for dealing with differences between props and jets are pretty transparent given all the other things you have to learn when transitioning to a new aircraft. I personally found the transition from prop to jet very easy, at least in the simulator training phase. I'll fly the real thing for the first time in a few days when IOE starts.