Shadowrun is
very different to D&D and Rifts.
As mentioned earlier, it's a game of eggshells armed with hammers. A starting character, with a lot of preparation and the right info, can kill a dragon. Not easily, but they can do it by the numbers. The thing that makes the greater difference is information and tactics. Treachery and planning beat raw numbers most of the time. And incidentally, dragons are very, very good at treachery and planning.

You don't level up. You never get more "hit points". You seldom wade through a room full of "low-level" mooks without caring because it only takes one of them to call HQ and then you've got Lone Star tailing you and it doesn't matter if you can beat up a Lone Star cop because what they'll actually be doing is following you back home and finding out where you live and sending SWAT teams at 3am. And even if you think you can wade through a room full of "low-level" mooks, it only takes one to have a grenade in a small room and you suddenly find out that the term "low-level" has no meaning in Shadowrun. Right place and time, with right skills. That's what Shadowrun is about.
To give you a couple of illustrations, in the introductory game I ran for my current group, they had to kidnap a man from a factory where he was hiding out. I gave them a "milk run", because it was our first time as a group. Well they easily outmatched the light security at the facility but they went in without any preparation, found the place was actually huge and ended up wandering from room to room looking for this guy. Meanwhile, alarm systems had been tripped, the target patched himself in to the camera systems, watched where they headed and nipped out via a different corridor. The only reason they knew he had left the place was because one of them had waited outside with the van and he promptly kidnapped this person to make them drive him to safety. How was all this different to D&D? Well for a start, when the target kidnapped the PC, there weren't rounds of combat wittling the PC down in hit points. Instead, the NPC passed a stealth roll, got behind the PC and put a blade at his throat. That would have given the NPC many bonus dice on his next attack if he wanted, very probably letting him kill the PC if he wished.
(The party pulled it back, by the way, stalling him with negotiation and then ending up in a Mexican standoff. The two groups eventually compromised with the NPC allowing himself to be captured, but only on condition of being handed over to the cops, instead of their employer. He figured with a good lawyer, he'd get off lightly. Which he did and is now a sort of contact of the group.)
Something else to watch out for is that whilst D&D hit points, healing surges and daily powers give you a sort of steady progression from top condition to dead without any diminishment of ability until that point and relatively easy estimation of how close you are. Usually players can say they're done for the day and make camp somewhere until they're all restored in the morning. In Shadowrun, you can find yourself running away very suddenly, and that running when you have a bullet in your thigh isn't much fun, either. And you'd better know where you're running to because in D&D the default setting is one where you have a culture that you can retreat back into and buy healing potions from. In Shadowrun, you are criminals and if Lone Star (the privatised police force) are after you, you might find checking into the nearest
inn coffin motel a little risky.
I hope that helps a little. I have a FAQ for new players geared slightly toward those coming from D&D backgrounds, it's
here. In fact, if you have a look through the Shadowrun section on my site (link in sig), you'll find quite a bit of information including some "NPC Rosters" and an "Opposition Roster" which are compendiums of grunts and pre-gen NPCs.
And, as others have said, welcome to the Shadows.

K.
EDIT: To clarify a little on the nature of progression in Shadowrun. It varies. You can start out quite maxed out in Shadowrun. You can't go "superhuman". If your character is already one of the strongest people around, then congratulations. You've reached the top! There's usually something to spend your karma on. It's worth being a well-rounded character and magical characters always have something good to spend their karma on. Mundanes tend to benefit more from financial rewards letting them buy better cyberware or equipment. Every team wants a van with a pop-out machine gun turret, after all.

But you also gain through building a better rep and getting better jobs as a result. My group are still living in a squat but after this next run, they'll be able to afford to get apartments of their own. It may seem a bit of a come down after D&D where you might buy a castle, but after living with the leaky ceiling and the devil rats, you learn to appreciate modest comforts. Shadowrun is a dystopia, after all.
And as to what attracts us to Shadowrun? It's gritty, it's dangerous, it's dripping with background and story and you have to think about who you piss off and if they know where you live. Or, as another Dumpshocker once put it: "Shadowrun is shooting people in the face for money. With elves."